Hello,
Well, actually, if you are referring to the JR East Tohoku Area Pass, there is a limit. You can ride as much as you want, but only on 5 days within a 14 day period. Of course the days do not have to be consecutive.
It really would be far more constructive if you stayed overnight in some of your destinations. From Tokyo to Shin-Aomori is 3 hours, meaning that you'd be spending 6 whole hours of the day sitting on trains - and that is on top of traveling at the start of winter, when the daylight hours are the shortest of the year. Tokyo to Akita is almost 4 hours one way and even worse. Yamagata is thankfully much shorter, taking about 2½ hours one way.
You can see regular fares, routes and schedules on Hyperdia.
As for within Tokyo, you can use it on the JR Yamanote Line and other JR lines in the city, although there is very little savings in it. If you carry out your plan, you would certainly get your money's worth and then some, but you appear to be forgetting that time is very important too.
Good luck.