You will have much better rates if you rent a Japanese SIM card. Renting a phone at the airport is a convenient option, and some companies will actually send you a phone by mail to your home or hotel. You will need to present a valid form of ID with a photo and a credit card. There are two sets of fees on rental phones, the daily rental fee (between 250 and 1000 yen) and the per-minute calling fee (between 70 and 200 yen per minute within Japan, no charge on incoming calls).
Prepaid phones which can be charged up at convenience stores and mobile phone shops, as well as subscription plans, are not available for visitors to Japan, you will need to be in possession of an alien registration card and/or Japanese drivers licence for that. Note that very few Japanese mobiles function outside of Japan due to the way they are setup and network conflicts.