You may get a credit card even if you are a foreigner in Japan, if you try for the "au Wallet credit card". (It's different from the "au Wallet card", without "credit".) I often hear that foreigners find it difficult to get their credit cards newly issued in Japan, even if they have multi-year permission to stay in Japan. I am not sure if that is really true. (For your information, it is very easy for normal Japanese people to have ones issued, unlike in the US. There, people need credit history, while in Japan, it is hardly important, at least for most Japanese people.)
One of my friends, who came from the US in July, 2014, was successful in getting a credit card with credit facilities at about seven thousand dollars, exactly seven hundred thousand yen, by application screening in just one business day and without a call to offices to check his enrollment. He is just a young ALT, assistant English teacher, at public schools so his income is not as high as foreigners who are working around Roppongi, Tokyo. He is a new graduate in the US so he has no credit history.
His card, with a VISA brand, without annual fee (permanently), and with a 1% rebate every purchase in what is called a "point program" commonly in Japan, is issued by au, the second largest cell phone company, for their contractors. It is possible that the fact that he has an au contract made application screening easier to some extent. However, I think that the most essential factor was that he told them that he is a government employee, 公務員. Unlike the US, being a government employee means that you have the highest credit capability in Japan. Again, that's different from the US.
Do you go to a dentist for periodical teeth cleaning? Do you know how to clean your teeth by using your health insurance in Japan, 健康保険? In Japan, just teeth cleaning by itself does not fall under health insurance. However, you can use health insurance for a combined treatment of both teeth cleaning and a medical checkup for periodontal disease and carious teeth. That combined treatment costs a total of about 3,500 yen, including one x-ray, and at least two times of going to the dentist. (You can't finish treatment going one time only.) The next time, after 6 months or more, you won't need to get this nominal check or x-ray.