Was in Tokyo last month.
I suggest going up to the topmost viewing platform of the Tokyo Tower at dusk and watch the city light up - beautiful views over the bay, Ropponghi & Akihabara. Not so much fun if there are still aftershocks, mind you.
For something traditionally cultural, the Meiji Shrine is a pretty typical example and is set in a pleasant park. Don't bother searching the park for the famous gatherings of local youth groups demonstrating their favourite music - they were banned in the Nineties and anything shown on TV these days has generally been set up for the cameras.
If you want to have a slightly creepy experience, visit the Yasakuni Shrine after seeing Meiji. The atmosphere of the place is a real window into the Japanese psyche (although many have problems with the war criminal associations).
If you want to see the famous pedestrian crossing that's shown on any TV programme wanting to demonstrate how crowded and busy Tokyo is, it's outside Shibuya station (I think). If you go up to the first floor of Starbucks and fight for possession of the window seat around 6-7pm (rush hour), that's quite a sight to look down over. And if you need a pint of Guinness to recover from the culture shock, there's a European-style bar called "Dubliners" about 100 yards up the road.
Remember that the Tokyo metro & bus systems shut down before midnight. Generally, the only people out after that time are the ones who can afford the £20 cab ride back to their hotel.
Great city - have fun!
Edited to add: If you want to visit the Imperial Palace Gardens, remember that they're shut Mondays, Fridays and public holidays. The Lonely Planet didn't mention that. Looked impressive from the outside though...
And if you're an aging rocker, don't forget to have your photo taken outside the Budokan.