Get yourself to Shibuya Crossing after sunset and watch the insanity unfold. A bazillion people crossing the street every few minutes, multiple screens pumping high-energy video from all directions at once. And more. It's the essence of Tokyo condensed into a single spot.
Go up to the Roppongi Hills Mori Tower top floor in late at night, around 10pm or so, and be mesmerized by the never-ending city scape and all the red, pulsing skyscraper lights. Seriously. Do it.
While you're at it, grab a caramel frappuchino at the Shibuya Crossing Starbucks, go upstairs and try and get a window seat facing the crossing. Should you want to experience some insane levels of process optimization and efficiency, go there during peak hours.
Get a front seat on the fully automatic Yurikamome line and head out from Shimbashi Station to Daiba station, on the manmade Odaiba island. Go for a walk along the shore. Great views of the Rainbow Bridge...
...and when you're close to Daiba station, make sure to catch the life-size Gundam statue in the park just behind the Diver City mall. It’s pretty awesome when lit up at night. The Diver City top floor has a pretty sweet fast food sushi place that you'd want to check out, too. Energetic J-Pop music mixed with iPad based menu systems delivering sushi to your seat with tiny automatic shinkansen trains.
Stroll through the Harajuku and Omotesando back alleys for a while. There's a loveley area just north-east of the Mejijingumae subway station. Enjoy the architecture of the various flagship stores lining the main street. Go inside Omotesando Hills. Grab an avocado burger at the Golden Brown place on the top floor.
Visit Yodobashi Camera in Akihabara. If there's anything you ever wanted in electronics, they're pretty sure to have it. Giant floors filled with cameras, cell phones, home electronics and a bazillion different accessories. Bikes? Yep. Toasters? Sure. About a thousand different headphone models? You bet.
Walk around the Tokyo Midtown shopping mall in Roppongi, grab a coffee at Café Idee Parc, then head up the Ritz Carlton lobby floor and have the keiseki lunch at Hinokizaka. Round off with today's cake selection in the lobby.
Find your way to Shinjuku and the Isetan department store, then go to the basement food floor. Be amazed.
Go visit the Tsutaya T-Site complex in Daikanyama. A prefect mix of books, magazines, music, film, a café and a lounge. A bookstore taken to perfection. It's only a few stations south of Shibuya.
Walk along the Ginza main street. There's a nice Apple Store, an insane Uniqlo flagship store and a ton of decent, upscale shopping along with quite a bit of interesting architecture, too.
In general, you really can't go wrong with food in Tokyo. It's really great almost everywhere. These are a few suggestions to get you started based on more than the food itself, and should you want to dig deeper – use bento.com. It's insanely useful when hunting for new places to eat in Tokyo. Or if you just want to find a useful list of animal cafés.
The Dentsu building in Shiodome has a collection of restaurants on floors 46 and 47. My favorite one is so/ra/si/o, featuring a sort of French-Japanese crossover cuisine. Stunning views of Tokyo Bay and the Rainbow Bridge. Lunch set menues are great value and a good way to get som food before boarding the Yurikamome line when heading to Odaiba.
If you're looking to get a classic, Japansese keiseki meal, Hinokizaka at the Ritz-Carlton is a pretty good bet. If you're a party of two, ask for the seats facing the windows and time your reservation to kick in half an hour or so before sunset. You'll be on the 42nd with stunning views of the Tokyo skyline. And yes, the food is awesome. Pieces of art, really.