score:4
Legally speaking, there's only one legal ID for foreigners in Japan: your passport. (Unless you're a long-term resident of Japan, in which case your zairyu card is also OK, but these aren't available to tourists.)
That said, ID checks are pretty rare and in practice you can likely get away with a laminated photocopy of your passport's photo page. Do bear in mind that, if the police decide this isn't good enough, they can and will detain you until somebody else can go get your actual passport and bring it up the police station, which can be quite a hassle to arrange.
For non-police ID use:
Nightclubs will insist on your passport, full stop. (Again, photocopy may or may not work.)
Hotels will insist on your passport. However, this can be trivially worked around by claiming to live in Japan and registering with a Japanese address, in which case they're not supposed to require it anymore. (No, that doesn't make any sense to me either, and not all hotels are aware of this.)
In most convenience stores, you need to self-certify that you've of age to buy alcohol, but there's essentially no enforcement beyond this and the staff generally are not paid enough to care. Same goes for most restaurants etc.
Alcohol and tobacco vending machines may require local "alcohol cards" (酒カード) or the nationwide Taspo card respectively.