How to approach a potential supervisor for "research students" in Japanese universities?
It’s normal to contact potential supervisors in advance to express your interest and they can support your application. It’s not uncommon to visit their laboratories or call by Skype with prospective international students. You can offer to meet them or give a departmental seminar, especially if you are attending a conference nearby and have travel funds. Many supervisors encourage this.

However, there are some things to be aware of. Japanese laboratories are highly hierarchical: lecturers or assistant professors do not have the same independence as in other countries. They typically defer to a more senior academic who runs the lab and may not have the power or funds to recruit students directly. If in doubt, you should contact the head of the laboratory (a tenured professor). Please be aware that they are busy including traveling to attend meetings and may be near retirement. It’s typical for candidates not to be contacted at all if they weren’t interested in taking you in Japan (this is the same for job applications). If they are interested, they should apply promptly.

The enrolment procedures in Japanese Universities are complicated and vary considerably between institutions. They can assist you with this but it understandable that some prefer to take students already enrolled in their institution. This includes research institutes which prefer to take students already enrolled in a Japanese University, although they can arrange to a host institution via collaborations. 

It is a huge commitment to host an international student. They should only take you on if they’re prepared to support you including the process to settle in to living in Japan and throughout your degree. In addition to the usual guidance required, an international student will also need help with the language and cultural barriers. They will also have to do a lot of paperwork in Japanese that you cannot. Not all Japanese laboratories are prepared to do this or take the risk on someone from an unknown institution in an unknown country. It’s best to introduce yourself clearly and point out any common interests or collaborators of theirs that you know. If anyone recommended to contact them, you should mention this. It helps to show initiative, show what you’ve already done to investigate your options and what you’re prepared to do (in terms of research techniques and funding your studies). Make sure you are eligible for funds if you are planning to apply for them to support your studies (and whether they will need to support your application formally).

If you can show a connection to Japan and that you’re serious about living in Japan with realistic expectations, that is helpful as well. There are genuine fears that international students will have poor manners or experience culture shock and abandon their studies to return home. You need to show that you are not such a “flight risk”.

They will also typically have stronger written English skills than spoken. As such they may not feel able to guide a candidate entirely in English. If you have any Japanese language skills or willingness to learn, you should show this. You should also try to show that you will share the burden of the language barrier, please use clear simple English to make sure they understand you. Avoid idioms, ambiguous terms, technical jargon, and of course: be polite and respectful.