Getting a credit card at a Japanese bank is rather difficult for foreigners, though I have heard of mixed success of getting a credit card from shopping centers such as Aeon or Amu Plaza/Yume Saito/Cocowalk. The people who sign you up for those cards work on commission, so while they do want you to get approved, be wary of the terms and conditions.
I personally applied for a Rakuten card online and the process goes something like this:
- Apply online (I had a JTE do it for/with me)
- Receive an email AND phone call in JAPANESE about approval and to check if you are a real person (you will confirm your birthday and other details over the phone)
- The card will be delivered by mail, and you WILL HAVE TO present ID
For Rakuten, we benefit by being JET Employees. We have a steady salary and can apply as government employees rather than contract workers like private ALTS. This gives you a better chance of getting the credit card. You can also earn points on their credit card to use on their site. I was approved for a roughly 200,000 Yen/ USD 2,000 limit my first time.
Here is a pretty comprehensive explanation of how the Rakuten credit cards work in Japan (payments and such).