Getting that permission is a matter of filling out a form and turning it in to the school, then waiting a few days.
Most of the students in Japan from places like the US don't usually do the typical part time jobs.. a lot of them are on one-year exchanges on the JASSO scholarship or something similar, which is for the most part enough to get by.
You might be able to have some success doing private English lessons under the table, or you might even find a small English school that will employ you part time. That is where you will make the most money.