Sweet, Japanese history. Fun topic.
I'm going to argue that the last Emperor to be the sole ruler of Japan was Emperor Go-Shirakawa, who ruled from 1155-1158. This could be a wrong assessment, as Emperor Shōwa(popularly known as Hirohito, r. 1926-1989) had a debatable amount of power during WWII. But I will try and justify it on the basis of three points:
In Modern Japan, political power has been shared between the Japanese parliament, the National Diet, and the Emperor, since the Meiji Constitution(referred to hereafter as 'the MC') of 1889. The MC granted a lot of power to the Emperor of Japan, and the powers given are similar to those of the British Crown: full command of the military, the power to dissolve/open parliament, and to sign off on laws. However, the Emperor's power is limited too: Article 5 states that the Emperor acts with 'the consent of the Diet'; Art. 8 gives the Emperor power in extreme circumstances, but gives the Diet the power to invalidate the Emperor's emergency actions when next in session; Art. 9 states 'no Ordinance[issued by the Emperor] shall in any way alter any of the existing laws'. These three articles severely limit the power of the Emperor, and I would argue, tip the balance in favour of the Diet in theory. In practice, however, it's hard to know how strong the influence of the Emperor is. Hirohito's rule in particular has been extensively studied by historians and several have argued that he exercised far more power than the MC allows. He is supposed to have achieved this by bypassing the government, and exerting control over the military directly, particularly during the war years. If this is the case, then he was in a way, the de facto ruler of Japan during its war years in the 20th century, but it would probably be more accurate to label him the Supreme Commander, as the National Diet maintained civil control.
Other Emperors shared power with the National Diet after the MC was passed. As discussed above, the MC created a system of government that did not give absolute power to the monarch, but did not abolish the monarchy either. Emperor Meiji, who ruled from the 'retirement' of the last shogun in 1868, also shared power until the MC in 1889. His power was shared with several ministers, and I would hazard a guess that the lion's share of the power was in the hands of those politicians, rather than those of the young Emperor(he was sixteen when he took the throne).
Prior to the Meiji Restoration of 1868, the Shoguns ruled the country for 683 years, give or take a few decades where the power was split between warring factions. The Shoguns kept the office of the Emperor alive as it was the source of power in Japan. Despite not being the actual rulers of Japan, the Emperors were there by divine right, and they were important to the power of the Shoguns. This thread goes into detail about that.
The most famous Shoguns were the Tokugawa line, who ruled during the Edo period from 1600-1868. Ieyasu, the first Tokugawa Shogun, had an interesting way of guaranteeing the succession of his family to the seat of power: he abdicated while still alive, and let his son Hidetada take the position of Shogun. Ieyasu retained his power, but the office of Shogun had officially passed on, guaranteeing that his son was legitimised and would hold power once he died. Ieyasu's son repeated the practice, abdicating 10 years before his death. This practice removed any succession issues at the death of a Shogun, and I bring it up because I think it is key to understanding where the power lay in the pre-Shogun era, when the last Emperors to hold control ruled.
The Emperors of Japan employed a similar strategy to maintain control and succession lines. The 'cloistered emperors' were Emperors who had abdicated their office and joined Buddhist monasteries, but remained the holders of true power while their successors were the figurehead. Emperor Go-Shirakawa, who I suggested was the last Emperor to hold true absolute power in Japan, ruled for three years as Emperor, but in reality it was slightly less as the former cloistered Emperor Toba did not die until the second year of his reign, in 1156. Go-Shirakawa's rule from 1156-58 was singular, and his abdication to join a monastery left his son Nijō to succeed. Starting there, the next four Emperors, and the final Emperors before the Shoguns took power, were Nijō, Rokujō, Takakura, and Antoku. Nijō was 15 when he became Emperor, and died at the age of 22. His abdication in favour of his infant son came after he fell ill, making it likely that he abdicated to ensure his son's succession. His son Rokujō was Emperor for three years, before being removed by his grandfather(Go-Shirikawa) in favour of his uncle Takakura, who was 7, which is a far more respectable age for an Emperor than 4. Takakura's abdication(closely followed by his death), made his son Antoku Emperor, at the age of 2. Antoku would die four years later in 1192, at the same time that the Kamakura Shogunate took control. Go-Shirakawa's time as cloistered Emperor ended in at the same time. During that period, he had ruled as two of his sons and two of his grandsons held the position of official Emperor. Two of his three years as Emperor were without a former living Emperor(1156-1158), and I would argue that those years were the last time an Emperor was the actual ruler of Japan.
tl;dr Emperors are cool, and baby emperors make for old Emperors running things. Hirohito may have run things, but only militarily, and Shoguns ruled the roost for seven centuries.