I believe that if you want to go to a Japanese Public High School you will need to speak Japanese, do you and what is your ability level?
The time span for a VISA depends on the type and what country you are coming from. I am thinking that you wouldn't be able to attend on a tourist VISA and would need to get a student VISA instead. To get one you will likely need to be enrolled in the school first and possibly have them sponsor your VISA.
From what my wife says the public schools do have costs associated with them but I believe they are mainly for uniforms, extra curricular activities, and meals if you choose to eat school lunch. I am sure someone here could give you better info though.
It sounds like a tough situation, I hope all turns out well.
Good luck Mark!