I agree with all the suggestions above.

You could still visit Mount Takao. It's less than an hour from Shinjuku, has, I think, the steepest cable car (a train on a cable) in Japan to get you half the way up, and is a lovely way to get into the mountains. At the top, you've got a decent chance at a view of the elusive mount Fuji. December weather is normally very clear and your chances of seeing Fuji are better than almost any other time of the year. You can always pop up the Tokyo Metropolitan buildings in Nishi Shijuku to gage your chances before you head off (you can go and come back within 5-6 hours). Mind you, I went to Takao at Xmas time last year and just saw white. Unusually, it had started to snow once we were half way up. Still had a lovely day though and will be returning for another attempt this year.

Chuckd is referring to the Odaiba part of Tokyo which rests on reclaimed land in the harbour. Tokyo International Exhibition Hall (AKA Tokyo Big Sight) is near Ariake Station (the end of the Yurikamome line). Palette Town is in front of Aomi Station (just a couple of stops away) and is probably the best point from which to start exploring this area. It's the newest part of Tokyo and is used by youg Tokyoites kind of in the same way that Coney Island has traditionally been used by young New Yorkers. And I stress 'kind of'.