I have been asked about this enough times to just write a blog about it. If I only had one day in Tokyo, this is how I would spend it (I grew up in Tokyo). And all of this can be done via the Subway system, which is super-extensive.
1) Start the day early at the Tsukiji Fish Market – you can see one of the world’s largest live Fish Market – There are some restrictions now like when you can visit – so check that out here You can get early morning Sushi breakfast, as early as 5am here. Its the freshest sushi you can get (straight from the ocean!).

2) Ginza: Take a morning walk/stroll from Tsukiji to Ginza, the Fifth Avenue of Tokyo – its a long walk, but you can see the heart of Tokyo come to life in the morning, plus you’ll walk by a classic Kabuki Theater.

3) Asakusa: Get on the Ginza line when you’ve seen enough and goto Asakusa. Asakusa has a big shrine and lots of little stores with Chotskies. Find some classic Soba noodles to slurp or Yaki-soba at a food stand.
4) Kappabashi: Take the Ginza line back toward the town center and get off at Tawara-machi. It is not very well-known, but where you find life-size plastic moldings of all kinds of foods, like sushi and ramen. It is used by restaurants thru Japan. It makes a great gift for back home.

5) Akihabara: Start your afternoon with a visit to Akihabara, he gadget/hobby/electronics heaven. You’ll find the entire neighborhood crawling with anime, latest gadgets, and weird electronics stores.
6) Harajuku: Take the Yamanote Line (above ground green train) Now assault your senses with the latest in Tokyo teen-fashion. This is where Tokyo teens dressed up goth before there was a word for it. There is a street called “Takenoko Doori” – literally Bamboo Street, that is a narrow winding street that has the highest vintage clothes store per square foot.
7) Meiji Shrine: Rest your senses with a serene visit to Tokyo’s largest shrine – Meiji was one of the great emperors of modern Japan.
8 ) Shibuya: if you can still walk, walk to Shibuya via Yoyogi Park (site of 1960 Olympics) its a good walk to the entertainment center of Tokyo. Visit Tokyu Hands, a hobby store that is now a 9 story department store with all kinds of little gadgets/tshotchkies.
9) Shinjuku: Take the Yamanote line to Shinjuku at night, and prepare to be “Lost in Translation”. Night scene (more specifically the safe red-light district) is here.

10) Roppongi: If you’re looking for English speaking bars and clubs, and for “late night” clubbing – you can’t beat Roppongi, which is where the gai-jin (foreigners) come for British pub crawl to hyper-kinetic disco dancing. It is close to the US embassy compound – which is why it became a “hip” part of town, way back in 1970s. You should easily be able to find something to do here until its morning.
It would be super-human to do this all in one-day, but if you’re not yet 30, perhaps you can.
Have a great time!