You may assume that Japan is a technologically advanced country and that you'll definitely have access to your online world 24/7 during your trip, but unfortunately, it's not an easy task to find Wi-Fi in Japan. However, with a little bit of preparation, the useless tablet or smartphone you have been carrying around can still turn into a wonderful door to the Internet.
I tried out Family Mart's free Wi-Fi service, which was good quality (fast and smooth). I'll first show you how to do the preparation in the store and then how you'll be able to connect to the Internet with some demonstration photos.
Note: They do not currently have an English page for the Family Mart Wi-Fi service, so I recommend that you follow the pictures carefully and in the correct order.
Thanks for the comment. As you said, if you don't have any WiFi connection, you won't be able to get the verification email. So what I recommend is to rent a phone with limited data and use it only when you want to make a new WiFi connection.