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There are a few places to get a SIM card for internet data access (without voice), but the prices vary widely, so be sure to shop around.
As of February 2015, the most affordable data plan was about 20 USD per week at Telecom Square, which you can find at both Terminal 1 and 2 (see their webpage for maps). Of course you need an unlocked GSM cell phone that you can change swap SIM cards. Additionally, some older cell phones are incompatible with the Japanese frequencies, but you could double check it at the store. You can add credit to your SIM card online if you plan on staying for a longer period of time. Despite what you may read elsewhere, you're not required to return your SIM card before you depart.
Alternatively, you can rent phones with data access, but it's much more expensive (about 10 USD per day) with the obvious burden that you need to return the phone before you depart.
You can also find other stores like Softbank in Terminal 2, but their prices are ridiculous (about 20 USD per day). Make sure to shop around.
Lastly, the airports have some vending machines offering data plans by Sony (so-net) / DOCOMO, but they are hard to find, and you have to pay cash.
SIM card with internet data access outside the airport:
To obtain a data plan at a cheaper price, head to an electronics store in downtown Tokyo such as Yodobashi Camera. They offer data only SIM card by various vendors, without throttling the bandwidth, and the price is better. A 1gb card for 3000 yen, 2gb for 4000 yen or 3gb for 5000 yen.
Japanese laws are strict about SIM cards with voice plans and basically require residency status. Your choice as a visitor is to use an app for making calls over the internet such as Skype or Google Hangouts. This turns out to be cheaper if you're renting a SIM card with internet access anyway. If you need a local Japanese number to receive calls, you can try creating one through one of those apps.