If you aren't lucky enough to have unlimited international data on your cell phone, one excellent option for you is to rent a pocket WIFI router for your stay. I have now done this several times and it made a huge difference to my ability to get around and stay in touch with people. There are a number of companies in Japan that do this - you have an option to pickup and drop off your wifi at the airport, or it can be delivered to your hotel/accommodation. I have only done the airport pickup/drop off and it was smooth each time. It also meant that I could check maps as I was on my way to the hotel:)
I added in the wifi rental when I booked my Japan Rail Pass. It ended up being Ninja Wifi, and I have since used them several times. Here is a review of your options in 2020, so you can make up your own mind.
Travelling is a lot easier with a cell phone. You can make arrangements to meet people, let them know you are running late, using GPS, check email...
You can make arrangements to rent a cell phone before you head off to Japan and have it delivered to wherever you are staying or pick it up at the airport. Likewise, you can drop it off at the airport before you fly home.
The Japan Guide has lots of information on companies and procedures.
Internet Cafes/Manga Kissaten are all over the place these days. They are usually open 24 hours and have "night packs" for people who decide to stay overnight instead of going to a hotel (or are homeless). Cleanliness can be a real issue - some are grungy smoky places. Others are bright and clean and safe.
Many public libraries also offer internet access. There is usually a sign-up sheet at the reference desk.