Another wonderful thing about convenience stores is that when it starts raining. they immediately roll a rack of cheap (¥500) plastic umbrellas onto the sidewalk for passersby who have been caught short.
Public buildings often have a plastic bag dispenser at the entrance. The idea is that you place it over your dripping wet umbrella, should you happen to have one. If there are no plastic bags, leave the umbrella in the rack at the door.
Not all ATMs take non-Japanese bank cards, even though ATMs were commonplace in Japan before they were commonplace in the U.S.
First Citibank ATMs became available in a few locations, and then, when Japan and South Korea co-hosted the World Cup, the post offices began offering international ATMs. Now they are available in 7-11 stores, and the Seven Bank ATMs in locations such as NRT also take foreign cards.
But you still can't walk up to any random Japanese ATM and expect it to accept your card.
You cannot buy a mobile phone as a non-resident foreigner. This used to be possible, but it became illegal at some point, supposedly because foreigners were using them to commit crimes (and Japanese people never did?). You can, however, rent a phone or a SIM card or rent a pocket wi-fi and use your phone's wi-fi function.