Also check Tokyu Stay or Weekly Mansion Tokyo. They are both business hotel chains and should be within your budget.

You can find affordable hotels in Shinjuku but they are some distance from the train station. For example, Tokyu Stay Nishi-Shinjuku is 20 to 25 minutes walk from the Shinjuku station by foot. Better take the subway.

Can't go wrong with Shinjuku. There's no shortage of things to do. There are plenty of department and electronic stores, Tokyo Metropolitan Government's free observation deck (vs paying at Tokyo Tower, Sunshine 60 or Yokohama's Landmark Tower), Meiji Temple, etc. Shinjuku is also a major transportation hub such that you can get to anywhere with minimum transfers.