Trying to think of activitites that don't involve shopping or eating (obviously the easiest way to spend time), visiting art museums came to my mind first. It's quite tricky here.. some museums I found don't have any information in English even on their websites.
I was happy to discover that there's Mori Art Museum on the 53rd floor of the Roppongi Hills shopping-leisure centre. They're holding an 10th year anniversary exhibition dedicated to love.
All You Need Is Love featured a wide range of artists from Jeff Koons, Frida Kahlo, Damien Hirst, Auguste Rodin, Marc Chagall and many contemporary Japanese artists including Yayoi Kusama.
Gimhongsok, Distorted Love. (After Robert Indiana's Love)
Jeff Koons, Sacred Heart (1994-2007)
Hatsune Miku (singing synthesizer application with a humanoid persona). The new form on linkage as seen in a virtual doll.
Damien Hirst. Untitled 2002.
Yayoi Kusama's installation Love Is Calling was the only artwork allowed to be photographed. Kusama's newly-comissioned (2013) work was a small mirrored room with odd colourful polkadot creatures. It wasn't as amazing as her Infinity Room I experienced in New York last summer but still cool!
Museum ticket also let you to go to roofdeck on the 60th floor. It was so foggy that you couldn't see very far... it felt kind of empty and creepy up there.