With some notable exceptions in hotel bars, happy hour is very rare for cocktail bars in Tokyo. However, there are some excellent happy hours with good cocktails to be found at restaurants throughout the city.
Cocktail bars generally do not run discount promotions or offer coupons. If you get in the good graces of a particular bartender/owner, there's a chance you could eventually score some free drinks here and there. But this is not something to count on or aim for.