Wrote this on another board, and I think I posted it here once, but this might prove handy as a general guide. I went to Tokyo Tower back in 2000 and was not impressed. Its a huge tourist trap, and I think the Tokyo Metropolitan Building probably has similar views with more to see and do in the area.
I would suggest the following, in no particular order:
Shinjuku (culture/shopping/nightlife) Its the main buisness district, and the municipal building has great views of the city. At night, the bordering Kabuki-cho area has hostess clubs and seedier places.
Shibuya (shopping) A trendy district where young people hang out.
Harajuku/Omotesando (shopping/culture) The trendiest area, super fashion forward, great people watching, and tons of tiny boutique stores with some larger flagship stores
Akihabara (tech/shopping) The electronics district with tons of multilevel stores- tech geek heaven. Don't go for good deals (Japanese goods are generally cheaper in the US), unless its something you can't find at home.
Ueno Park (culture) Nice park, and right next to an impressive museum of Japanese artifacts.
Tokyo Edo Museum (culture) Fascinating look at Japan's history - great exhibits.
Ginza (shopping/culture) The fifth ave/Rodeo Drive of Tokyo, and the Kabuki theater performances have in-ear English explanations.
Roppongi (shopping/nightlife) The bar/hostess club area. Worth a gander if that's your thing. Nearby is the newish Roppongi Hills for upscale shopping/bars
Imperial Palace (culture) great views on the outside, interesting gardens on the inside.
Tsukiji Fish Market (food/culture) go early morning for fish auctions, eat the freshest sushi around.
I know thats a lot, but those are only the major tourist attractions! If time is short figure out the type of sites that are priorities, and plan accordingly. You can easily hit 2-3 areas in a day if you plan well.
Hope this helps!