Regular PASMO, Suica and ICOCA transit cards are available again at all train station ticket machines, but if you buy a temporary visitor transit card when you arrive at the airport, it won’t cost you the ¥500 non-refundable deposit and can be reloaded for 28 days from the date of purchase.
OR
If you have an iPhone (sorry, Android users) you can get the mobile Suica app, which can be reloaded for 180 days (from date of purchase) from the credit card in your Apply wallet. Here’s how. It also avoids that ¥500 fee. Here’s how to use the mobile app on your phone to ride the trains.
Where to buy visitor transit cards at Haneda Airport:
They are sold on the Arrivals Level (2nd floor) at the JR East Travel Center and at the dedicated “Welcome” card ticket machine nearby. If you follow the airport signs to the monorail, you should end up here:
The vending machine only takes cash, so if you don’t already have a pile of yen, get some from the 7-11 ATM right next to the ticket machines. (Note: 7-11 ATMs accept foreign-issued debit and credit cards—unlike most Japanese bank ATMs—and are the easiest and best way to get cash, wherever you are in Japan.)
• Choose your language
• It will then give you several choices of how much to load onto your card to get started. Pick one.
• Feed yen notes into the slot in the middle (any denomination, doesn’t matter which way you put it in)
• It’ll give you a “processing” message, then spit out your Welcome card, your change, and a receipt that tells you when the card will expire and can no longer be reloaded.
More useful info:
Practical Japan travel advice (using the trains, how to travel in Japan with food allergies, where to get cash, and stuff like that) is at Travel Tips and Tricks
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And here are the other places I take my friends when they come to town
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Jonelle Patrick writes novels set in Japan, produces the monthly e-magazine Japanagram, and blogs at Only In Japan and The Tokyo Guide I Wish I’d Had
Jonelle Patrick writes about Japan 
Thanks so much for the tips for the Suica card at Haneda Airport – pictures and instructions absolutely spot on. Saved us a lot of stress late at night when we landed :-)
You totally made my day! Nothing makes me happier than knowing I helped you get off to a great start in Japan! Have a wonderful time, and I hope it’s even better than you dreamed ^_^