Let's say the professor had a lecture abroad and met a very keen and promising student there. He agreed on giving him a position as a student intern in his research at a university laboratory. It is completely outside of any publicly open positions but the prof seems to be very impressed by the student's smart questions and interest. However, apparently, it's the university who has the final say whether they will officially invite him or not. The university is Japanese. What are the factors upon which they make the decision? And what are the odds they will agree to that? I am sorry if this question is off. It's torturous not to know if there is any hope or not.
Different schools have different procedures, but as far as I know, a formal offer of admission will come from an admission officer (or a graduate officer), not from a professor.
Usually, the most a professor can do is to recommend a student for admission to a program, and possibly indicate that she/he has already met the student. The admission process itself is usually handled by a separate officer, which will check if all the admissions requirements are met. Those might include various academic requirements, suitable background in the primary or a cognate field, language requirements etc. (These requirements might be checked at the departmental level, and a recommendation then forwarded to the appropriate admission officer.)
In some cases, admission may depend on the availability of funding, so that a student who applies knowing a professor can guarantee funding has an advantage.
Funding is usually the trickiest part, but I am also aware of cases where a student had found a willing supervisor but the file was rejected on academic grounds.