as far as I know you can work with a dependent visa. I have a friends who's husband is an engineer and she came her on a dependent visa and she works as a teacher in a kindergarten school. And until now she has still a dependent visa and she has no problem whatsoever. I don't think you need to apply for a part-time visa, and I don't think there is such a visa
But if you are a degree holder, you could apply for a working visa once you get here, just be sure to bring your credentials. And check out Japans' Ministry of Foreign Affairs site for more information.