For new graduates, you submit your information through their website and receive an account for future application-related news/correspondence. Next step is submitting more information and answering a couple generic questions (Why Rakuten?, etc). After that is an IQ type test with three sections: English, Math, and Reasoning (with diagrams). I found the diagram test to be pretty difficult. Finally, there is a one-to-one interview (through Skype for those outside of Japan). This whole process spans a couple of weeks.
If you are in Japan, you might be able to attend a hiring seminar about the company. Otherwise, there will likely be an online video to watch.
I think it's probably best to speak Japanese at the interview, if you can. Otherwise you might have to talk to an unattractive, young American man who has at least JLPT N1 ability. You will be asked a lot of generic interview questions. You should be knowledgable about web services and the like.
According to the American interviewer, he was part of one of the first large groups of foreigners to enter the company, starting last year. Also, may be of interest, he graduated from USC.
On their CV form they ask about your college activities and clubs. And, as all standard Japanese resumes tend to require, you will need to include a recent picture of yourself. Otherwise, nothing remarkably different about the hiring process verses a large American company.