How long will you stay? There's plenty to see in Tokyo. The basic touristy stuff is the Meiji Shrine, the Empiral Gardens, and the sort. I highly recommend checking out a few sumo matches. Although, like mentioned, Fuji-yama will be cold in December (I climbed it at night in late September and it was plenty cold then at the summit), you can still visit it. You can still climb it if you're prepared and experienced (it's an endurance hike rather than a technical one). It can only be seen from Tokyo on a clear day, so it's worth spending a couple days around its base; lots of hotels, restaurants, and the beautiful lakes.
Japan is a lovely country. Enjoy! Be sure to let us know how it all went.