Tokyo has hundreds of museums and art galleries. Many of the most famous such as the Tokyo National Museum and the National Museum of Western Art are located in Ueno Park, but there are many more spread throughout the city. Entrance fees for museums are typically ¥600-800 and ¥1,000-1,500 for art galleries, with discounts for children and senior citizens.
If you plan on visiting several museums in Tokyo, then you should consider the Grutto Pass which for ¥2,000 gives you discounted entry to 70 facilities throughout the capital. Note that most museums close on Mondays (unless it is a public holiday), and during the end year and New Year holidays.
For a listings of current and upcoming art exhibitions in Tokyo see the Art & Museum Exhibition event page.