While it is technically possible to complete in one day, you're going to be pretty tired and at least ¥10000 poorer by the end, so splitting this up into bite-sized portions is advisable. Lots of detours from the main itinerary are provided, pick and choose the ones that sound interesting.
Due to its length and complexity this is not really suitable as a layover tour. If you have less than half a day to spare, you're better off sticking to the Ueno or Asakusa districts, both within easy reach of Narita airport — see Classic Tokyo, Modern Tokyo for a sample route through Asakusa.
There's a lot of travel involved, so a ¥2000 Suica or PASMO fare card (available at any train station) will let you zip around the city easily without needing to queue up for tickets at every stop. You will need to pay an additional 500 yen deposit when you first get the card. It does not matter if you get PASMO or Suica, they are interchangeable on all lines in the Tokyo, Chiba and Yokohama area. All private and JR lines take PASMO and Suica cards.
Start off your day bright and early with a visit to the Tsukiji Fish Market. Tourists access to the tuna auction is limited (no auction on Sundays). If you just want to see the market in action it is much cheaper to just take the first train (underground) in the morning a little past 5AM to Oedo Line Tsukiji-seijo, or Hibiya Line Tsukiji, but this is a little further away. Be sure to eat a sushi breakfast at Sushidai or Daiwa Sushi for around ¥4,000. If you just want to eat, you can show up a little later, but beware: queues can be long on weekends.
Tsukiji Honganji (築地本願寺), near the Hibiya line station, is a rather atypical Japanese temple built not from wood, but from heavy stone and concrete. The interior, full of wafting incense and resplendent with gold, is still worth a quick peek.
Hop on the Toei Oedo Line to Ryogoku and the Edo-Tokyo Museum (exit A3/A4), one of the best in Tokyo, which will give you an excellent grounding in the history of this city from the Edo era of samurais and geishas to modern-day postwar Tokyo. Admission ¥600, open from 9 AM, closed Mondays.
Right next door to the museum is the Kokugikan, Japan's most famous sumo wrestling arena, where tournaments are held three times yearly. A visit to one of the sumo stables nearby can be interesting, but must be arranged well in advance.
If you're feeling geeky, you can stop just a few stations away at Akihabara and plunge into one of the world's largest electronics retail districts. You can also find oodles of software, games, comics and various mixes of the two here. Remember that most everything here is aimed squarely at the Japanese market, so voltage (for hardware) and operating systems (for software) may not be compatible, and the language in the manuals certainly won't be — check out the export retailers like Laox for international versions, but be prepared to pay an "international premium" as well.
Ride all the way through central Tokyo to Yoyogi, then change to the Yamanote Line and ride one stop south to Harajuku. Immediately to the west side of the station is the majestic Meiji Shrine, one of the largest and most serene in Tokyo, located down a wide foot path into a forest of tall cedar trees. Once at the shrine entrance, rinse your hands and take a sip of cleansing water before entering. (Note: Do not drink the water. Take the dipper in your right hand and pour water over your left hand. Change hands and pour water over your right hand. Change hands again and pour water into your cupped left hand, transfer the water to your mouth, rinse and spit---yes, spit--out the water into the trough at the foot of the fountain. Again, rinse your left hand, rinse the dipper to clean it, then put the dipper back on the rack.) Here you can make a wish (remember to throw a coin; a five-yen coin is preferable) into the money box as an offering. Also, notice the other worshipers bow and clap twice to call the gods or buy a votive plaque (ema) to write a wish on. If it's a weekend outside winter, the odds of catching an elaborate Japanese wedding ceremony here are pretty good.
On Sundays only, there's a ceremony of a different sort going on outside the shrine entrance and in nearby Yoyogi Park when the unofficial Tokyo freak show is held. Here you can catch punks, gothic lolitas, bloodspattered surgeons and other bizarre subcultures showing off to each other and the gaggle of photographers.
Backtrack to Harajuku station and cross to the east side, where an entirely different vista will present itself: right across the road is Takeshita-dori, the nexus of Tokyo's teens, home to the world's heaviest concentration of Hello Kitty goods and other forms of extreme cuteness. Kawaiiiiiiiiii!
Walk through the narrow winding street and take a right at its end onto Meiji-dori. The next intersection is Omote-sando, a tree-lined boulevard occasionally compared to Paris' Champs-Elysées, with trendy boutiques and snooty cafes priced almost as high as the original.
Feeling a little peckish? Stop at Tenya on Meiji-dori (on the right side before the Omote-sando crossing) for a ¥500 bowl of tempura and rice (天丼 tendon).
Cross Omote-sando and keep walking past the Condomania shop. A few blocks down, take a right to cross under the Yamanote train tracks, and after this you are now officially in Shibuya, Japan's capital of cool. The shops here change at a blistering pace nearly as fickle as Tokyo teen fashion, but a few long-termers along the road include the OIOI (say "marooee") fashion mall and the seven-story Tower Records music and book store, where foreign (read: English) books can be found on the seventh floor.
At the end of the road you will find Shibuya station and Hachiko, the busiest pedestrian crossing in the world. If you want to explore Shibuya a little more, make a sharp right here to stroll down the pedestrian street Center-gai. Along the street one shop that's worth a stop is Tokyu Hands, a DIY department store that retails absolutely everything imaginable (and some things that aren't), and otaku certainly won't want to miss out on manga/anime superstore Mandarake.
When night starts to fall, board the Metro Ginza line to Shinbashi and change to the Yurikamome line to the artificial island of Odaiba. This futuristic all-automated train-bus-monorail is an attraction in itself, especially the approach to Odaiba via a 270-degree loop that propels the train onto the Rainbow Bridge. There's lots of futuristic architecture here, including the spectacularly bizarre Fuji TV building, and even a copy of the Statue of Liberty by the seaside.
If your quota of shopping still isn't full, detour from Odaiba-Kaihin-Koen station to the mindboggling Venus Fort, a recreation of Venice inside a shopping mall, complete with artificial sky and Italian mayors giving speeches from balconies.
After all that sightseeing it's time to take a dip. Get off at the Telecom Center station and cross the parking lot to Oedo Onsen Monogatari, Tokyo's spiffiest spa complex done up to look like the good old Edo days. Get a locker key, pick a yukata bathrobe of your choice and change into it, then head out into the spa armed just with the key. There are restaurants, bars, souvenir shops and various Edo-era amusements, all of which can be paid with your key. Entry after 6 PM costs ¥1900.
Re-energized, you can hop on the Yurikamome and ride back to Shiodome. Change here for the Toei Oedo line to Roppongi. If you haven't had dinner yet, take exit 4 and walk a few hundred meters to Roppongi Hills, where you will find countless eating and shopping options in superslick surroundings. Try the tonkatsu (deep-friend pork cutlet) at Katsukobo Wako (NB1F) or the curry udon at Konaya (NB2F).
Wash down dinner with a beer or seven in one of Roppongi's innumerable watering holes. These change with bewildering rapidity, but Gas Panic and Lexington Queen have been around forever. For a more upmarket clubby experience, check out Space Lab Yellow, another reliably quirky standby.
When morning comes, stagger onto the first Oedo line train to Tsukiji for a sushi breakfast, and start the tour again!