Great question! I study colonial Mexico and my answer focuses on contact between Japan and Mexico (and Europe) in the early 17th century. Hopefully others can add more on Japanese knowledge for an earlier period (updated from earlier answers:

The Philippines had been conquered by the Spanish in 1571 and since then formed an important trade link with New Spain (colonial Mexico). According to Serge Gruzinski

Ever since the Jesuits had set food in the archipelago, Japan had attracted numerous Iberian missionaries and merchants. The establishment of the Spanish in the Philippines had brought Asia closer to Mexico, and made Japan a focal point for royal officials, merchants and monks, all of whom saw it as a providential springboard for the fabulous land of China.

The execution of six Franciscan friars in Japan 1597 (later made martyrs by the Catholic church) became quickly known in Mexico and came as a shock. This event in tandem with Jesuit reports that the highly developed Japanese were the “Spaniards of China” led to an increasing fascination with the region, not only in Europe but also in New Spain.

This was affected by other events – including the first two Japanese diplomatic missions traveling to Mexico, and the second one from there on to Spain and Italy. They made stops in Mexico City in 1610 and 1614 before Japan’s increasing policy of seclusion from the 1620s onwards.
Birgit Tremml-Werner goes into their historical context:

Tokugawa foreign policies are conventionally considered as significant improvement for Japan’s international relations. When [the shogun, Tokugawa] Ieyasu took the reins in foreign affairs after winning the Battle of Sekigahara (1600), he at once got in contact with neighbouring countries. For this reason, post-1598 foreign policies have often been regarded as completely different from Toyotomi Hideyoshi’s aggressive stance. The number of international negotiating partners and letters exchanged with foreign rulers increased dramatically during the first years of Tokugawa rule. ...

Soon after taking up leadership, he took steps to comfort the Spaniards in Manila after several years of distress. Aware of the benefits of long-distance trade, Ieyasu soon targeted the Castilians of Luzon for collaboration on a trans-national level. Herein lies the major difference with Ming China. Early modern Japanese pluralistic Bakufu aspirations even included a desire to integrate Japan into trade with Mexico. As far as the Manila trade was concerned Ieyasu’s goodwill politics remained ambivalent and were hard to match with the already complex official foreign relations portfolio.

The Spanish Crown however was increasingly opposed to Japan building ships for trade with Mexico and possibly other American regions - this would have meant a clear rival for their trans-Pacific trade and monopoly. So that Spain did not allow for any such maritime or trade collaborations, even before Japan’s stopping most foreign influence from the 1520s.

In both cases, the missions went from Japan to western Mexico across the Pacific. The first group then went overland to Mexico City, and later returned the same way. The second group traveled from there to Veracruz; and then to the Spanish court in Madrid and to the Vatican - roughly the direction of the Manila Galleon (Manila-Mexico-Seville). This route was established by that time but not yet very secure, including the dangerous winds. The very first Japanese delegation to Europe - so even before those to Mexico - in 1582 had traveled via the older route, with a stopover in India.

A bit more on the 1st delegation to Mexico: Rodrigo de Vivero, a nephew of New Spanish Viceroy Luis de Velasco, had made an emergency landing in Japan in 1609 during the crossing from the Philippines to New Spain due to a storm (showing the dangers of that crossing). There he met the shogun Togukawa Ieyasu.

According to the Aztec historian Domingo de Chimalpahin (see below), the Japanese ruler had borrowed several thousand pesos to the Spaniard Rodrigo de Vivero, so he could set up some trade there. Vivero returned to Mexico City the next year with the Japanese delegation, officially to improve relations between Japan and the Spanish Crown - but also to settle his own debts. They went on the San Buenaventura, supposedly a Spanish ship, which landed in Matanchén (in today's state of Nayarit).

The 2nd embassy in turn was not sent by the shogun but by a prominent daimyo, Date Masamume, who resided in Sendai and ruled a large area in northeast Japan. Date's ambassador Hasekura Tsunenaga, the crew and other Spanish officers as well as 150 Japanese left Sendai on 27th October 1613, heading to Acapulco.

The ambassador Hasekura Tsunenaga of the 2nd dipolmatic mission traveled from Japan to Acapulco and met with the New Spanish viceroy. In Mexico City, Hasekura offered the New Spanish government free commerce between the New Spanish territories and Japan, with one of his main objective an opening of trade opportunities. From there the mission traveled on to the Spanish court where its leader was baptized; and from there later on to Rome to meet the Pope - as this wonderful picture testifies, plus this one in slightly lo fi.

While not much resulted from these missions in terms of economic or diplomatic exchange, they did further raise interest in Japan. We also know that some members of the missions stayed on in or near Mexico City. The major Nahua (Aztec) historian Domingo de Chimalpahin, living at that time and place, tells us how some members were baptized and stayed on for a few years to work as merchants - including at least one Samurai though that's another story [edit] which I added here due to popular demand!

Most Japanese seem to have returned to Japan by the 1620s due to the changing Japanese domestic policies; but contact had been made. After Japan was opened up economically centuries later, from the later 19th c. on economic exchange with Mexico was taken up again in a different form.

I should mention that today there are statues of Hasekura Tsunenaga in Acapulco, Cuba, Spain, Italy, the Philippines and Japan, testifying to a renewed interest in this exceptional journey.

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So: Japan in the late 16th/ early 17th century clearly had knowledge of and attempted to open trade relations both with the Spanish Philippines and Mexico, with one diplomatic mission even crossing from Mexico on to Europe. These were short-lived, with the increasing isolation policies after shogun Ieyasu from the 1620s onwards. Still, contact had been made in an increasingly interconnected early modern world. By this time, all then known continents were already connected through trade, colonisation and ecological exchange - and not all those links moved directly through Europe.

cont'd below

This happened towards the end of the Japanese feudal era, but it's still a fascinating story.

3 sailors from Japan were shipwrecked in the Makah Nation, on the northwesternmost tip of what is now Washington State. It's the first known contact between this region and Japan. They ended up living with the tribe for a time, then they went to Vancouver, then London and then China. https://www.historylink.org/File/9074

If you go to the Makah Museum (and you should), you can see a Japanese boat that was given to the tribe in the modern era as a gift of thanks for keeping these castaways alive.