1. Convenience Stores
The main convenience stores like 7-Eleven and Family Mart all have free Wi-Fi access in English in most of their stores. Just open up your browser in the store, click the English button and follow the instructions. Usually just providing your email address is enough to access the internet. (For reference, you can see an explanation of how to use Lawson free Wi-Fi here.)
2. Train Stations
Many train stations, in particular Tokyo Metro (subway), have free Wi-Fi. JR (national rail) stations usually aren't so good, but there are typically convenience stores right outside these anyway. Just open up your browser, and if the station has free Wi-Fi, you'll be redirected to a registration page. Just providing an email address is usually all that’s needed.