Other than that, Tokyo isn't all that interesting for tourists. Sure, there are some things to do (a decent museum, gawking at the skyscrapers in government-office part of Shinjuku, gawking at the lit-up streets at night in the nightlife section of Shinjuku, etc.), but nothing really on the must-do list.
I would disagree with this. I think it is like saying that all there is to do in New York is to go to the Statue of Liberty, Empire of State Building and hang out in times square or at the Moma. Tokyo is a mega-city...it would require a lifetime to exhaust the possibilities of things to do or see, and by the time you did, there would be countless new things to do. In that way, it is like New York, London, Beijing, Moscow, Sao Paulo or any of the other giant cities in the world.
Don't get me wrong, I agree that it is very much worthwhile to travel outside Tokyo; I lived for six months in Sapporo, a minor city, and I travelled all over Tohoku and Hokkaido. That said, I think it is incorrect to imply that Tokyo is only about skyscrapers and neon lights. It is a very new city though...we destroyed the old city in the fire bombing in 1945, essentially flattening the entire city. They built it from the ground up in the 50s and 60s. If you are looking for old Japan, Kyoto or Nara are better options, but that does not mean that Tokyo is of no interest to tourists -- it really depends on what kind of tourist you are. Anyway, I wish I could offer more concrete advice, but I have only spent two weeks or so in Tokyo, spread out over several visits, so I do not know it as well as Sapporo, for example. That said, I always had a good time just wandering around in different neighborhoods -- there is always so much going on on the streets that it is worth setting out for a specific neighborhood and seeing what it brings you. Shiodome is a lot of fun, and quite different from Shinjuku, which is different from Akihabara, which is different from places like Chiyoda.
If you are interested in history (or even Asian current affairs), places like the Yasukuni shrine and its museum are fascinating as they offer a totally different view (and pretty much universally discredited) of world war II -- one where the Americans were the aggressors and where Japan did no wrong in China. It is a window on why the Chinese and Koreans were so upset when Koizumi went there to pray every year.
Or, there is Nikolai-do -- a Russian Orthodox cathedral built by Nikolai Kasatkin, who later became St. Nicholas of Japan. I did my graduate work on him, so I am biased, but he was a fascinating man. He was sent to Japan alone on missionary work and to found a Russian orthodox church there...alone among Western missionaries, he learned the language fluently and conducted services in Japanese, translated the bible into Japanese and gained great traction among the locals. I am agnostic myself, but his life was fascinating. He built a Russian Orthodox cathedral in Surugadai that is still around today and rather interesting (to me anyway).
I am not saying you have to go to these places, but I offer them as examples of what is out there if you look for it. What kind of things interest you? Because if you do research beforehand, you should be able to find some amazing things in Tokyo that relate to your interests. Obviously it is chock full of camera gear and photo galleries, it is mecca if you are a record collector, and there is a tremendous music scene. The fish market at Tsukiji is amazing, and if you are interested in architecture a place like Shiodome will be fascinating. Overall it is a very interesting city. I recommend you pick up a Time Out guide for Tokyo...they seem to do excellent guides in my experience. It might help you out.
Shiodome
And the elevated walkways there:
Shinjuku outside yodobashi camera...I am sure you will go here!
Also shinjuku
Sorry, I don't have many pictures from there...I did not spend as much time there as I would have liked.