I would suggest going to Harajuku. It is a very interesting part of Tokyo with the younger people's culture of dressing up in wild outfits. There is also some good shopping nearby. Also, near Shinjuku, is the Meiji Temple (er, perhaps Shrine). Very, very beautiful. In central Tokyo, you can see the Royal Palace. You can't go inside; but, it is an impressive site (it sits on a large very green hill and looks like a huge green space in the middle of skyscrapers). If you can, get Lonely Planet's Tokyo Guide. It is really helpful. I don't have it to hand at the moment or I could tell you more (having trouble remembering the names of places atm).
Knowing a bit of Japanese is helpful. When I was on my own, I made sure to attempt to speak what I could. Simple things like: "Hello," "Goodbye," "Thank you," etc.
Perhaps Madeira will pop on and tell you a bit more. She lives in Japan now.
Congrats on winning the trip! Japan is an amazing place and I am sure that you will have a great time.