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.
These three companies are the major cell phone providers in Japan:
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.
The apps are amazing. Tokyo subway apps, navigating apps. Yokai (demon) apps. Dictionaries.... But, if you take your iPhone to Japan, remember you need to get an international plan and be careful about how you use it so you don't end up paying an arm and a leg.
In order to buy apps from iTunes Japan, you need to have a Japanese credit card and an address in Japan. You can also use an iTunes card to get around the credit card, but you still need to provide an address in Japan.