If you don't mind an early start, Tsukiji Fish Market is an amazing place. If it's the right time of year, you can even see the tuna market, where each fish is worth more money than a reasonable house.
The best lookout in Tokyo is free at the Tokyo Municipal Government Building. On a good day, you can see Mt Fuji!
Meiji Jingu is a small piece of quiet reflection in the middle of the hustle and bustle.
Shibuya is the "cool" district, and has so much to see and do, from cool bars and shops, to the statue of Hachiko, the loyal dog at the train station.
We even walked across the famed Shibuya Crossing. Like predicted, we are big in Japan.