My suggestion is to avoid bars around the East exit of Shinjuku. There have been cases of a girl or a guy sitting with customers and helping them to order drinks, hide are really expensive as the customers discover at the time of paying.

The Shinjuku ward office is aware of the problem, and has responded by opening a tourist information center. The, you can find the names of reliable and reputed bards with price-marked menus for all drinks. Support in multiple foreign languages is available.

@Tanuki's answer is an important one to note, there has been many incidents like these in Shinjuku area.

There are other small scams that I have seen, but these are minor:

1. There are some people who are hanging around the entrance of Ueno Park and raising funds for earthquake victims. Usually, fund raisers get a permit from the police and stay at one location. But these people stay within the crowd, and have been collecting for ages. I am not sure if they are genuine or not.
2. In busy stations of Tokyo, a well-dressed middle aged person might tell you that he lost hist wallet in the train and ask you for money to get home. Japanese police boxes lend money for people in such cases; this guy just targets foreigners who are more likely unaware of this. I recently noted some foreigners (looked like they are from China) playing the same trick on other foreigners.