I lived in Japan for a year.
Tokyo is bullshit. If you want to go to a big city where you can do a bunch of shopping and stuff sure, but it's not much more than a huge busy city. You might as well go to New York. It's also the most expensive area of Japan.
I recommend the Kansai area. Osaka is a fucking great city that has a bunch more culture and it's a big city with some cool clubs and stuff. Shinsaibashi and the Amemura area in Osaka is great. About 30 mins from Osaka is Kyoto which is a city older than most civilizations. It's a bit touristy but some cool sights. Bunch of temples and shrines around too, make sure you check out Nara, some cool temples and you can buy swords and throwing stars outside Todaiji for less than $30.
As for cost, if you know what you're doing, it can be freakin cheap. There are restaurants that can be pretty damn expensive but if you want to do it cheap and cheerful you can hit up yoshinoya or ramen houses and get masses of food for less than $2. You should also try yakiniku, all you can eat barbeque meat for 80 minutes that usually costs $20 a head. They give you a barbeque pit in the middle of your table and shitloads of raw meat you just throw on the fire then eat. The food will be quite different, so prepare yourself for that (it's not like chinese).
Try to avoid drinking at bars, it's stupidly expensive considering beer and cigerette vending machines and convience stores can be found almost everywhere. You can buy huge cans of beer for 200yen at vending machines, 7/11s and Lawsons as opposed to drinking a tiny glass of beer at a bar that costs 400yen.
Finally make sure you go to a karaoke box. You get a small room where you can get around 6 friends singing "we are the champions" or any other song you can think of (they have a surprisingly massive library) at the top of your voices while you take advantage of the nomihoudai (unlimited all you can drink alcohol) you get at most of these places. They literally have a phone on the wall and you call them for more and more drinks. These session can for a few hours at a time, it's best to start around midnight and finish around 5am when the trains all start up again. It's pretty damn cheap too.
All in all get around, explore, get lost and meet some interesting Japanese people who are willing to take you around the cool spots.