Yes, is it possible, in a rather roundabout way - That largely depends on your age and how lenient the school is. If they`re sucking up to the public offices then they probably won`t do it, but it is possible.
Personally, I`m wondering why exactly you`re limiting yourself to "public" schools. Public high schools in Japan aren`t anywhere even close to being free, so if it`s a money issue there is no difference between public and private. Private schools are much more lenient and will be more likely to work with you.
There is an odd category called GAP. I`m not exactly what it stands for, but it`s not a student and not a teacher. Basically you are a volunteer at the school in English classes, and in return the school volunteers to let you sit in on other classes. As no money is exchanged, the visa problem never arises, and there isn`t exactly a time limit on it.
I found out about this by "volunteering" to help at a local high school in hopes of making friends. I ended up joining the English class for 2 hours a day, and free to visit other willing classes and clubs during the rest of the school day.
If you have any other questions please ask me.
