As for Tokyo itself, if you're looking for a hostel I always recommend K's House in Asakusa. Awesome staff, less than a minute away from Kuramae subway station, and very close to all Asakusa sights.
I never saw much of Tokyo itself, but from what I did see...
Avoid Roppongi by all means for a night out.
Best night view of the city I saw was up Tokyo Tower. I tried the free one at the government building but the fact that the lights were on bright just spoilt the scene from the reflections in the windows.
Do not miss Shibuya Crossing at night! It's alot better than Times Square
Enjoy!
