I've never been to Nagoya, so I can't tell you what the public transportation is like. I'm sure you can contact the school and maybe they can put you in touch with someone who can give you that info..maybe like a penpal?
They do have those sorts of soaps there, but not sure what you would use being allergic to those brands. I'll be there's someone that can answer that question for you also!
It's not a silly worry...it's something you'll need to get figured out.