The 90 day Temporary Visitor Visa doesn't allow you to work. I believe you might be restricted on study options as well. Some countries have an arrangement that allows you to extend the 90 day visa from within Japan. USA isn't one of them though.
You are best contacting the Japanese embassy where you live to find out what visa's you might be eligible for and how to apply for them. Usually to work you'd need a job offer from a company that will sponsor you, and to study an offer from an educational institute, then a certificate of eligibility is issued from the Ministry of Justice in Japan. Some countries have working holiday agreements with Japan.
http://www.mofa.go.jp/j_info/visit/visa/index.html
I know people work on a temp visa but the work you'd be able to find would likely be low paid with zero benefits. Japan's economy has a bit of a problem at the moment (and foreign workers were amongst the first hit) so I'd guess the job market is fairly competitive and the spare cash to spend on English lessons might not be in great supply.
If you are of independent means and can prove this to an entry officer you just need to leave the country and re-enter again.
I've never had such close entries and exits and I might be in an age bracket they don't examine closely but since they introduced photos and fingerprints the entry officers don't even speak to me. Customs officers still do 'though.