Suntory Museum of Art
Housed in a modern Japanese building designed by architect Kengo Kuma, the Suntory Musuem of Art is a striking white structure with vertical louvers that run the length of the exterior. The design sets the scene for a high-concept museum that displays "art in life" including ceramics, laquerware, textiles, and glass from ancient to modern times. The museum includes a tea ceremony room that lets visitors experience this Japanese tradition the second Thursday of each month. There's also a cafe and gift store that sells functional art.
While in the 'art triangle' district of Roppongi, check out the Mori Art Museum and National Art Center: there are discounts for visiting multiple museums.