It’s been a couple years since I’ve visited, but I wouldn’t think things would have changed all that much.
Buying postcards was pretty easy - they seemed to be as available as they are in the US. All the tourist attractions had them, and most of the hotel gift shops had them.
As to mailing them, I would simply take them to the hotel desk, and ask them to send them (to the US). They would tell me how much for postage, and that was that - pretty simple. In some cases they’d actually sell me stamps and point me to the mailbox. But most times they’d just take the cards and handle it from there.
(All this without speaking a word of japanese).