I've just been on a 3 week trip to Japan and didn't see traditional Japanese dates used anywhere except on museum exhibits. Train tickets, bus tickets, hotel check-in / out dates, credit card receipts, tickets for various attractions etc. all displayed the date in ISO-8601 format.
It's interesting to ponder, but the traditional date format seems to have fallen out of use (probably with the advent of the computer era) and I genuinely don't think that this will be a problem.
Probably not with the tourist , though all my JR ticket I have are listed with traditional date though -- maybe they issued it with A.D because of you are tourist. Other tickets seems to be mixed though.
For example, of documents I have laying around my desk right now:
Alien card: A.D
/ Student card: A.D.
/ My number card (like US SSN): Traditional
/ Insurance card: Traditional
/ Water bill: Traditional
/ Electricity bill: Traditional (this one doesn't even tell you that it's traditional, it just say 30-07-xx)
/ Monthly health insurance bill: Traditional
/ Gas bill: A.D.
JR definitely produces different tickets depending on whether you used the machine in English or Japanese. (Or depending on what the station attendant thought would be most convenient for you; they can pick the output format and I've gotten both.)
I lived in japan for a few months and all the dates for the immigration, housing, etc... in official papers had to be written in the traditional format.
You'll see more in banks and sometimes registering papers and I admit I hate it because I don't remember it as I use it maybe only once or twice a year and it provides no benefit to anyone.
It's interesting to ponder, but the traditional date format seems to have fallen out of use (probably with the advent of the computer era) and I genuinely don't think that this will be a problem.