If you visit only one museum in Tokyo, make it the Tokyo National Museum. Here you'll find the world's largest collection of Japanese art, including ancient pottery, Buddhist sculptures, samurai swords, colourful ukiyo-e (woodblock prints), gorgeous kimonos and much, much more. Visitors with only a couple of hours to spare should focus on the Honkan (Japanese Gallery) and the enchanting Gallery of Hōryū-ji Treasures, which displays masks, scrolls and gilt Buddhas from Hōryū-ji (in Nara Prefecture, dating from 607).
With more time, you can explore the three-storied Tōyōkan (Gallery of Asian Art), with its collection of Buddhist sculpture from around Asia and delicate Chinese ceramics. The Heiseikan, accessed via a passage on the 1st floor of the Honkan, houses the Japanese Archaeological Gallery, full of pottery, talismans and articles of daily life from Japan's prehistoric periods.
Check whether it's possible to access the usually off-limits garden, which includes several vintage teahouses; it opens to the public from mid-March to mid-April and from late October to early December.
The museum regularly hosts temporary exhibitions (which cost extra); these can be fantastic, but sometimes lack the English signage found throughout the rest of the museum.