Well, interestingly, many people associate the advent of firearms in Japan with the Meiji Restoration in the 1860s and modernization to European standards because that has this really stark contrast in the popular imagination of feudal samurai, these guys wielding katanas and wearing the fierce looking men-yoroi, leading the Japanese equivalent of the charge of the light bridge against lines of imperial soldiers wielding rifled firearms and wearing european style uniforms, but it's not when guns first appeared or had widespread use in Japan. Firearms emerged in a big way at the end of the Sengoku era, starting in the mid 1500s and exploding in use, particularly under Oda Nobunaga who famously loved guns. It only (sort of) fell out of use in the Edo era under the Tokugawas, although guns were still manufactured in Japan, because it was just an exceptionally peaceful era, what conflict happened was small scale and local. The gun resurged as part of the modernization of Japan under the Meiji Restoration's military reforms and modernization which gave us those really stark contrasts many assume were the first times guns were used in Japan.
There is a whole class of firearms called Tanegashima, the matchlock harquebuses the Japanese developed from Portuguese and (later) Dutch imports. Their name indicates their origin: Tanegashima is one of the only islands the Japanese allowed foreign trade to enter, it was specifically reserved for Portuguese traders (it's similar to how the Japanese called much of western science "Rangaku" (蘭学) - Dutch studies - because it came through the Dutch port in Dejima). If you google image search Tanegashima, you'll alternately get images of beautifully crafted matchlocks, and also just maps and photos of the island.
The Japanese adopted the firearm toward the end of the Sengoku in the middle of the 16th century. For those unfamiliar, the Sengoku was a period of incredible, endless warfare in Japan starting from the 15th century after the power of the Shogunate was broken in the Onin war, and only ending when a new Shogunate formed under Tokugawa Ieyasu whose military supremacy was unchallengeable. The country fractured into smaller feudal states and their clients in almost endless war with each other over two centuries. It was in this period that most depictions of the itinerant samurai warrior emerge, as previously most samurai were aristocratic and tightly tied to a lord/daimyo and his land, but in this period many peasants became de facto samurai, because that's just kind of how war works (those same warriors at the end of the Sengoku would evolve into the bureaucrats, poets, and artists of the peaceful Edo period as the Bakufu).
The most famous example of this fluidity is actually the final reunification of Japan under the Shogunate, which happened basically in stages over 3 families: Oda Nobunaga, who did much of the foundational work for reunification, Toyotomi Hideyoshi, who by hook and crook came into control of the Oda clan after Nobunaga's assassination, and finally Tokugawa Ieyasu, who kind of picked up the pieces after Hideyoshi failed to cement his dynasty. Nobunaga and Ieyasu were the scions of powerful clans, but Hideyoshi was a peasant - his biography doesn't even give him a surname before he just started receiving/taking them as part of distinguishing himself as a competent soldier, negotiator, and general under Nobunaga.
Oda Nobunaga was a big fan of tanegashima and used them extensively in his reunification. Much of his armies were peasants/conscripts (ashigaru), as one might guess from Hideyoshi's humble background and rise to prominence, and many were armed with tanegashima, in contrast to the professional/aristocratic samurai. By the time Hideyoshi invaded Korea in the 1580s/90s using the armies he and Nobunaga had built, it had tens of thousands of arquebusiers in a force of 160,000, an exceptional growth in use when the first arquebus to arrive in Japan was only 50 years earlier in 1543. Guns continued to be manufactured in Japan in the Edo period even though the country was closed to foreign trade, they were just pitifully outdated by the 19th century. While the gun fell out of favor in such a peaceful era, in contrast, the bow developed a ceremonial purpose by becoming integrated as a meditative practice in Zen buddhism via Kyudou ( 弓道 ) - the way/path/dao of the bow. So under the bakufu/Shogunate's government, the bow found a place of cultural relevance even in a time of peace, but the tanegashima never quite found a niche in the cultural museum after its time of high prominence at the end of the Sengoku.
Noel Perrin wrote an interesting book on this, Giving up the Gun: Japan's Reversion to the Sword, which, while it has problems, includes a lot of information. Other sources:
Tanegashima: the arrival of Europe in Japan, some of my old textbooks whose names I can't remember.
EDIT: as /u/ParallelPain pointed out, I misquoted the number of arquebusiers sent to Korea. It was a significant portion but the total force was 160,000.
A few other members have done a fantastic job answering the question so far, but as I was well into typing my response to the OPs question, I will post regardless to provide some additional context. Apologies for any overlap.
To start, I will lay out an overview to go over the general overviews of the topic, before diving in more deeply. Firearms replaced bows in Japan during the mid 1500s to 1600, when at this point it was evident that the former had supplanted the latter. In 1543, Japan was introduced to western style firearms. Throughout the preceding decades (leading up to 1600) adoption and implementation of firearms was gradual and inconsistent. Many different factors (such as location, relationship to bakufu or Portuguese, etc) means that certain areas of the country had more availability and understanding of the weapon. Coupled with this, if we consider the temperment of the daimyō of a domain regarding the implementation of firearms, it was not a meteoric revolution that immediately changed the whole face of battlefields in Japan. Firearms did surpass bows in use, but it was a gradual and inconsistent process.[2] Now to dive a bit deeper let's look at a short narrative of events surrounding firearms in Japan leading up to 1600.
As I stated earlier, European firearms were introduced to Japan by the Portuguese in 1543, but this was not the first time Japan had seen (or used) firearms as a whole. The Chinese for some time had their own version of firearms. These were brought to Japan through the Kyūkyū Kingdom (modern day Okinawa) about 80 years before the Portuguese introduced western firearms. A version of these primitive firearms which were introduced via Ryūkyū were are three metal tubed weapon which was called hiya, fire arrows. Archeological records, dairies, and military account all indicate that hiyas were brought, and used throughout the different regions of Japan. While they were there, they did not see widespread use in the slightest. ([3] pg. 146) In 1543, 3 Portuguese merchants happened to land in Japan when the Chinese ship they were on was blown off course due to a storm. ([1] pg. 203) They landed on a small island 44 miles to the southeast of Kyūshū, called Tanegashima. The firearms (teppōs) were demonstrated to the residents of Tanegashima, and the Portuguese eventually gave 3 of their guns away. 2 of the firearms were given to Tanegashima Tokitada, who was the local lord, and a third was bought by a monk of the Negoro-ji temple. From this event, teppōs would spread throughout Japan. ([4]pg. 143-144)
The monk returned to the firearms back to Negoro-ji temple in central Japan. Their the monks used the metal forgery they had ties with to start producing teppōs. The gunsmiths created enough firearms to arm a unit of 300 teppō shū (firearm wielding marksmen) in 1570. ([3]pg. 124). In Tanegashima, Portuguese merchant ships arrived two years after the initial contact in 1543, and a trade relationship was established. ([1] pg. 264) Tanegashima brought teppōs to the Ashikaga shoguns, along with recipes for gunpowder. The Ashikaga shoguns would proceed to give out teppōs and the gunpowder recipes to lords who would support them. Japanese gunsmiths did not take long to become proficient in the production of teppōs. Along with the aforementioned forges in Tanegashima and Negoro-ji, forgres producing firearms popped up throughout Japan, such as in Sakai. ([4]pg. 156)
Now moving forward onto the adoption and implementation of firearms on Japanese battlefields. Much has been made as to the now semi-mythical relationship between Oda Nobunaga and firearms (which has been challenged to a degree, such as with the reconstruction and push against more traditional understandings of the Battle of Nagashino in 1575), but some of the earliest successful use of firearms were by the previously mentioned force of 300 teppō armed monks of Negoro-ji. Thomas Conlan discusses in Instruments of Change when he states:
“In spite of Negoroji’s proficiency in using and producing these weapons (firearms), their role in disseminating firearms has been ignored. Standard narratives of Japan’s sixteenth-century history portray regional “lords” or daimyō as being the most cognizant of the power of these new weapons and most able to use them effectively. Oda Nobunaga, the first of the “three unifiers” of Japan, has been characterized as a military genius whose concentrated use of firepower allowed him to “revolutionize” warfare, crush his most potent rival, the Takeda of Kai province, and consolidate power from 1570 until his assassination in 1583. The priests of Negoroji realized the importance of these new weapons earlier than any daimyō. In contrast to Nobunaga, who hastily assembled a squad of gunners for the battle of Nagashino, they fielded a formidable squad of gunners through mastery of production and training. And when Nobunaga dispatched his brother to attack Negoroji, his army suffered a convincing defeat. Unlike the “epochal” encounter at Nagashino, this battle has been consigned to oblivion, largely because Nobunaga’s successor, Toyotomi Hideyoshi, incinerated most of the temple complex in 1585.” ([3]pg. 124)
This is an important passage to highlight for several reasons. One it highlights the flaws in the more popular narrative surrounding the adoption of firearms in Japan. This usually is something along the lines of ‘guns were brought to Japan in 1540s, were not widely used until Nobunaga came along, and he saw the advantage in firearms and was able to seize control of Japan because of it, and as a result other warlords saw the advantage of firearms’. In actuality, the spread of firearms was very inconsistent, from both a time and location perspective. Warriors of Kyūshū and western Japan saw a larger influx of firearms early on as opposed to the warriors of eastern Japan. ([4] pg. 179) Even within regions, to which degree and how quickly proper firearm usage was implemented varies between Japan. If we look at three major samurai houses of the east, the Uesugi, Takeda of Kai, and Hōjō, this condition is apparent. The Uesugi leveraged their close relation with the Muromachi bakufu to acquire teppōs and gunpowder recipes. The Uesugi worked on methods of organizing gunners to fire in groups, which provided them an advantage earlier than their counterparts. Mobilization reports indicate that the Hōjō were slow in figuring out how to properly implement firearms in their military structure, and that it was not until around 1587 that they were able to reach a 1:1 ratio between bows and firearms. The Uesugi were much faster in this regard, and as Conlan notes in Weapons and Fighting Techniques of the Samurai that Uesugi Kenshin was able to have significant success vs the Hōjō in a campaign in 1560-61. Prior to this campaign in 1559 he had received a recipe for gunpowder from Kyōtō. To what extent this affected the campaign could be debated, but it is worth considering. The Takeda of Kai were relatively slow in implementing firearms as well. Documents from 1562 indicate that the Takeda planned organize their troops in units of 45 soldiers, with only 1 of them being a gunner (comparatively, there would be 5 archers). ([4]pg.165) These three warrior houses give a good example on how varied the implementation and use of firearms were even within a given region. Different groups took varying amount of time to properly implement firearms. Eventually daimyō were able to integrate teppō gunners into their military structure to a degree that in 1600, for the most part, firearms had surpassed bows in usage in Japan.
u/Khenghis_Ghan has already given a lot of background about the widespread adoption of firearms (the matchlock arquebus/musket) in Japan, so I won't repeat any of that.
Japan did have earlier exposure to gun (Chinese guns), but made little use of them. The introduction of the Portuguese-style arquebus, usually dated 1543, resulted in the widespread use of guns (see Brown (1948) for an overview of pre-1543 guns in Japan). The Battle of Nagashino (1575) is often described as the first major battle where guns were decisive; approximately 10% of Oda Nobunaga's force (the winners) were equipped with arquebuses. So, we have about 30 years from the introduction of this type of gun to it playing a key role on the battlefield. Not counting pre-1543 guns, this is a rapid process. The speed at which these guns spread and were used in increasing numbers on the battlefield resulted from a combination of factors:
The Portuguese-style arquebus was a mature and effective design.
Japanese metallurgy and metal-working was sufficiently advanced so that such guns could be made quickly and cheaply, and of good quality.
It was a period of very active warfare in Japan, with regional warlords contending for greater power, or attempting to survive in the face of neighbours with growing power. Warfare was characterised by increasing size of armies, and greater dependence on infantry. A weapon which (a) offered much better armour penetration than the bow, and (b) required less training for effective use than the bow (not needing the physical development required by high draw weight war bows), and could be obtained in large numbers sufficiently cheaply was attractive.
By the time of Hideyoshi's invasion of Korea in 1592, the gun was not only important on the battlefield, but also dominant. However, the bow didn't disappear from the battlefield; in at least some of the regional contingents that went to Korea, the number of archers was about the same as the number of gunners. Experience in Korea led to an even bigger emphasis on guns, with requests from the Japanese forces from Korea like:
Prepare guns and ammunition. We have absolutely no use for spears. It is vital that you arrange somehow to obtain a number of guns. You should see to it that those persons being deployed [to Korea] understand this situation. The arrangements for guns should receive your closest attention.
(quoted in Stavros (2013)). Even after the ascendance of the Tokugawa shogunate at the Battle of Sekigahara (1600) and their crushing of their major remaining opponents at the Siege of Osaka (1614 and 1615), writers such as Miyamoto Musashi consider the bow to still be usable on the battlefield In his Book of Five Rings, he compared the bow and the gun:
The bow is tactically strong at the commencement of battle, especially battles on a moor, as it is possible to shoot quickly from among the spearmen. However, it is unsatisfactory in sieges, or when the enemy is more than forty yards away. For this reason there are nowadays few traditional schools of archery. There is little use nowadays for this kind of skill.
From inside fortifications, the gun has no equal among weapons. It is the supreme weapon on the field before the ranks clash, but once swords are crossed the gun becomes useless.
One of the virtues of the bow is that you can see the arrows in flight and correct your aim accordingly, whereas gunshot cannot be seen. You must appreciate the importance of this.
If major warfare between powerful warlords had continued, it's likely that the bow would have completely disappeared from the battlefield. However, with unification of Japan, and the stable Tokugawa government maintaining peace, further military development largely stopped, and the bow was still present in small numbers (the gun - the matchlock arquebus - was still the dominant battlefield weapon) in the mid-19th century when encounters with the West (Perry's fleet, and all that) convinced that Japanese that modernisation of their armed forces was necessary.
References:
Brown, Delmer M. “The Impact of Firearms on Japanese Warfare, 1543-98.” The Far Eastern Quarterly, vol. 7, no. 3, 1948, pp. 236–253. JSTOR, www.jstor.org/stable/2048846 doi:10.2307/2048846
Miyamoto Musashi (c. 1645), Go Rin No Sho (Book Of Five Rings). The quote is from the Victor Harris translation.
Matthew Stavros (2013) Military Revolution in Early Modern Japan, Japanese Studies, 33:3, 243-261, DOI: 10.1080/10371397.2013.831733