You can work 20 hours a week on a pre-college student visa, but you still have to have the $20,000USD in the bank before they'll issue you the visa.
A cheaper option would be to study on a Working Holiday Visa. For that one you only have to show $2,000USD in the bank. The working holiday visa is issued initially for a 6 month term which can be renewed for another 6 months.
I've never done an exchange program so I don't know what the requirements are for it. I've also never done WHV (American's can't do WHV), but I have several friends who are here studying on WHV right now.
Here is official information on the Working Holiday Visa along with requirements:
MOFA: The Working Holiday Programmes in Japan
Also keep in mind how much it costs to live in Japan. I live in Tokyo and it costs me about $1,500 a month just to live (rent, food, train pass, but not including tuition costs). Many part time jobs pay something in the area of $10 an hour. That's about $800 a month. If you can find a teaching job, you might be able to make more money. But realistically, you need to be able to cover atleast half of your living expenses without relying on work income. Which means you need about $8,000 or more in the bank for living expenses.