I found that a prepaid phone (with AU) was ridiculously expensive and inconvenient. However, before getting my visa that's what I had to use.
If you are on a visitor visa, you can get a prepaid phone with your passport and cash. When I first came to Japan I didn't have a job, so that doesn't matter because you just pay there and it's over.
If you're on a long term visa, you have many more options. I've only really dealt with AU, but once I had my gaijin card and proof of a long term visa (written on your gaijin card) I could select any phone I wanted. I can pay with an American credit or debit card too. If you don't want to use a card, then you can just pay at any convenience store or even the phone store.
In any case, I would definetly reccomend having a cell phone number on your resume. Some employers won't even consider you if you have no phone number.