score:5
As far as I am aware there is no specific passenger rights legislation in either country. However, the airline is contractually obliged to either get you to your destination or refund your fare (although computing this may be complicated for a half-used return ticket), and in Japan in particular, airlines will generally bend over backwards to make things right.
I would just call them and see what they have to offer. If you can offer a specific remedy, like booking your return on a different airline that has a suitable flight on the right day, or flying you somewhere else and paying for the taxi, it would likely make the conversation easier.
The technical term for flying another airline is "endorsing" you onto the other flight, and while your ticket is almost certainly marked NONENDORSABLE, airlines can and will make exceptions when the issue is their own fault. Endorsing within the same alliance, in your case likely Asiana to ANA, is particularly common, but airlines can endorse with pretty much anybody if they want to.