This sounds like a great first trip! Japan was my first time abroad when I was 19 and, having never really been outside of Oregon, I found it to be quite a shock. It's now 12 years later and I try to visit every couple of years. Tokyo is truly a magnificent city. Here's my advice re the hotel:

Having lots of spending money is imperative in Tokyo, more so than almost any other location. Since you are only going for two weeks, I would suggest spending less on your hotel and saving the extra for sightseeing and, more importantly, trying as much Japanese food as possible. I spent a summer at the Asia Center of Japan which ranges from 6,000-10,000 per night, depending on the type of room you want. See: http://www.asiacenter.or.jp/
I stayed in the single room without a private bath and thought it was just fine for the money. The rooms are tiny, even more so than other rooms in Japan. However, they are clean and comparatively cheap. This hotel is also only a short (read cheap) subway ride from Shibuya and Harajuku, a couple of the cooler areas in town for young people. It's also walking distance to Roppongi, which may or may not be your scene (lots of drunken American military guys). There are lots of similarly priced business hotels out there - I usually use this one because it's easy to get home cheaply if I miss the last train.

I would also highly recommend trying to make a few Japanese contacts before hitting the ground there. Tokyo is a huge city and I found it difficult to meet local Japanese people outside of the school or work context. Having a local there to take you to an izakaya or a great local restaurant will make your trip much more memorable.

Edited to correct (large) error in price.