For cultural stuff, go back to Kyoto. Hands down. Spend a weekend in Kyoto if you can, it's well worth it, and probably better than Tokyo if you want to see traditional Japan. Do the Hozugawa boat trip. I was supposed to do it the last couple days before I left Japan, but it was cancelled due to a typhoon; I'm still sour about that, bigtime.
I second the Odaiba/Minato-ku/Akasaka area; one of the nicest parts of Tokyo IMO, and sitting out by the Rainbow Bridge at night (take the Yurikamome monorail) is just stunning. If I could live anywhere in Tokyo, it would be Minato-ku with a view of the rainbow bridge.
Akihabara for anime/manga crap.
Roppongi has lots of meat markets as far as clubs go, and Roppongi in general gets a bad rap as lots of military people go there and lots of foreigners, etc; however if you get OFF the main strip in Roppongi, Roppongi is really, really nice and quite a very wealthy area with lots of high end places tucked in the woodwork. Roppongi hills is good for shopping.
Make time for Meiji shrine, it's off the Yamanote loop line; I think it's right outside either Harajuku station or Yoyogi station, but I don't recall for sure. Also make time for Senso-ji, which is in Asakusa; very famous temple with a corridor of trinkets leading up to it, beautiful old part of Tokyo. Shinbashi is also the old part of Tokyo, and is gorgeous, stroll through there.
Ginza for high end shopping, and the most expensive real estate in the world.
Harajuku to see Japanese freaks/goths. Definitely a neat place in its own way.
A good general foreigner pub is the Hub, there are quite a few and usually close to train stations in popular areas.
Do karaoke, I know it sounds retarded, but it's all you can drink and you'll be surprised how fun it can be even if you hate singing like me; find a few ladies to go up to the karaoke box with and it's even more fun.
Tsukiji fish market is supposed to be fantastic, though I've never been. Go at the crack of dawn when they open when everything comes off the boats.
If you want a great Japanese eating experience, look up a place called Yakitori alley, it may or may not be in Lonely Planet; it's a string of open air yakitori (skewered grilled chicken) shops literally under the tracks between Ginza and Shinbashi. There are quite a few open air yakitori shops there with little wood stools and tables made out of stacked up milk crates and lots of traditional red hanging lanterns, and the smoke rises up from beneath the tracks...there is nothing like going out for yakitori and beer at a little open air shop, there just is no comparison Stateside. This is a must do. Find this place, you'll know it when you get there.
You can probably pass on seeing Tokyo Tower, no big deal; you can see it from afar from the Rainbow Bridge anyway and that's good enough.
Kabuki-cho in Shinjuku if you want a good look at Japan's underworld, filled with yakuza (mafia) etc. Very cool place. If you prefer the company of men, Shinjuku ni-chome is the place to go.
Definitely, and this too is a must--go to a yakiniku (beef) restaurant while you are there. Hands down the food I miss the most from Japan, day in day out. Absolutely hate that I have to go 400 miles to Chicago to get anything close. I second izakaya, these little Japanese pubs are great. Again, Japanese pub food and beer, can't beat it.
Shinjuku and Shibuya are the places to go for clubs/youth/nightlife, both on the Yamanote line. Spend your weekend nights here. Hachiko crossing in Shibuya is basically Japan's Times Square, right outside Shibuya station, west exit I believe--look for the huge building that says QFRONT on it with a starbucks in it, that's Hachiko crossing. LOTS of bars and clubs in that area. Stay out of sleazy Gas Panic in Shibuya, Sam and Dave's can be good if you want to meet girls who like foreigners, but the best club I remember going to was Camelot which is also in Shibuya, but a hair off the main strip; I stumbled across it and had never heard of it and there was a line around the block, about 70% women when I went, so I just chatted up a few and got in line with them and had easily one of the best nights in Tokyo. After four drunken hours in there having one of the best times of my life, I only then realized there were actually two floors, what a mind**** that was. Ageha and Womb are must dos as well, though Womb is damn near impossible to find. Womb is more for techno/trance/DJ type stuff, but it's big and dark and disorienting and tons of fun. Camelot/Ageha/Womb are the places to go IMO for Tokyo clubs, but most anywhere is good and if it sucks just go somewhere else. I really wouldn't spend much time in Roppongi clubs, there are better places to go.
Also Japan has no dress codes which is nice. You can also drink on the streets and in public, no problem; I liked grabbing a couple beers from the conbini (convenience store) and people watching outside while we drank a bit, and then moving on to a small pub to get a good buzz going and chat up some locals, and then moving on to the clubs late at night. Also there are no closing times, a club here and there may say on the door they close at 12, they don't. Saying they do is just some municipal ordinance, but it's not uncommon for 2 AM to be the hour in the club where it really picks up and people drink til dawn. Don't be surprised if the clubs still aren't in full swing at midnight, they will be.
Also for what it's worth, Saturday is a noticeably larger night for going out than Friday night, not that Friday isn't a great night too.
God I miss the hell out of that place now.