Please read the FAQs. This has been answered there (by me).
No, you do not need a degree to work here. It is the most common way, of course, but you don't need a degree for the following situations:
1) student visa. With special permission, you can work part-time.
2) cultural visa. With special permission, you can work part-time.
3) working holiday visa. No sponsor needed. Aussies can get this for 18 months.
4) dependent visa. With special permission, you can work part-time. You must be married to a non-Japanese who has a proper visa.
5) spouse visa. No sponsor needed, other than your Japanese spouse.
6) regular work visa. SURPRISE! Yes, you can get this with no degree whatsoever, but you must prove a certain number of years of related work experience. 3 years for teaching. An employer is needed to sponsor the visa.