Nowadays, there are various options to have a mobile phone in Japan, such as signing a new contract with a major cell phone carrier, buying a low-cost SIM, or renting a mobile Wi-Fi.
This page introduces some service providers as examples. Of course, you can use another service as well.
Mobile Service Providers for Non-Japanese Residents
Low-Cost Mobile Service Providers
Major Mobile Phone Carriers in Japan
You can find instructions on how to sign up for a cell phone in English language in the webpages below.