1. For the jet-lagged: You'll be up at 5 a.m. anyway, so why not hop in a cab and head over to Tsukiji fish market? Witness everything from gigantic frozen tuna being sliced with an electric saw to buckets of tiny live eels. Motorized carts take a cornucopia of seafood through the seemingly endless warren of stalls, but if you're careful and polite, the fishmongers won't mind your gawking and may even offer you a fresh fishcake.
2. Sumo is hugely (pardon the pun) popular in Japan, with matches held in giant arenas. For a behind-the-scenes look into this world of brute strength and surprising agility, visit a sumo beya, or stable, for an early morning practice session. In a steamy room with tatami mats on the floor, sliding paper doors, and no furniture, you'll sit just a few feet from the grunting and sweating wrestlers as they go through a series of group exercises and practice matches in a dirt ring. The sight is awesome -- and free. No flash pictures, but do bring a treat for the wrestlers.
3. If your business in Tokyo didn't allow any free time, don't worry about coming home empty-handed. Arriving at Narita airport with the requisite two hours before your flight, you'll find great little shops selling souvenirs at prices comparable with those of department stores. Check out the kimono-style cotton bathrobes, samurai headbands, and fancy snacks.