I believe that you have no understanding of what a Minor League General Manager's duties are. They have nothing to do with the players. The Major League affiliate handles that. They're involved in running the business end of the franchise, such as hiring office personnel, ushers, ticket sellers and ticket takers, concession stand and souvenir stand workers, etc.
You have to start out on the lower end of the totem pole, doing one of the lesser jobs, or even interning.
My suggestion would be to contact a Minor League club near you, and have a talk with the General Manager, to get a good idea of what college courses to take, and how to get started.
I worked with a club in the Class A California League for about 14 years, and saw how they operate.
Good luck.
Sarge