As an ALT with an instrutor’s visa. It is almost a must to do part-time work. When recruiter’s we're ALTs, we also had part-time jobs as well.
If you are in an instructir’s visa, you need to go to the immigration office and receive a sticker in your passport and a stamp on your residence card that allows you to engage in other activities other than your specified visa. You must take the contract of the company you will work for, as well as a permission letter from your current company to the immigration office to engage in other activities . It is simple process and you can get them on the same day at the immigration office.
You are not allowed to work at bars or restaurant not places that are close to your school.
Popular part-time work is English conversation schools on the weekends but make sure it does not interfere with ALT work. Online teaching is growing as well. Online teaching does not require a stamp or permission so feel free to do this type of work in your down-time.