Typically, when a clerk needs an ID for a transaction, they ask for a passport. In most cases, a drivers license will be accepted. I can't tell you specifically about trains, but I don't know why they would need a passport.
Don't you want a one-year extension of stay on the visa? That's much easier than looking for short-term gimmicks, and then starting over (I assume) with a new visa application.
I didn’t say that you need an agency. I said it’s worthwhile, because trying to keep track of the constantly-shifting procedures is practically a full-time job— and for someone who only needs to do it once a year, it can be an exasperating waste of time.
You should be able to qualify for a one-year extension of stay based on retirement, not OA but O, you can apply without leaving the country. Proving you have steady income might be a little complicated.
Foreigners are supposed to be reported by every place they spend the night. So technically that’s your responsibility. But unless it’s a long stay, or you’re hosting people as a business, I don’t think anybody really cares. But if you and your landlord were going to be complying with that rule, you would need passport numbers, not phone numbers.