Ill give what information i know, as i tried to enquire regarding Universities in Japan etc so know some information, but just wanted to make a point as a observation from what iv read and come to understand regarding this particular career and Japanese Culture.
In short, Psychology might be a very hard career to pursue in Japan. From what iv read, seen from peoples experiences (from the japanchannel.com videos also) is that how Japanese society works and how people act and react within the society is they generally keep alot of personal thoughts and emotions to themselves and they never express these outright to others. Generally how the society is 'conditioned' (if thats a good word to use?) regarding expressing thoughts and feelings it might make having a career in psychology very tough. I would say there are few people doing this in the country
On the otherside though, because of this and the fact the Government has recognised the figures concerning suicides etc they have/might put more focus towards this and as a result a market for psychology may be there. But i actually haven't seen many College/Uni courses in Japan for Psychology, if any at all. No harm in looking though.
Correct me if im wrong on any information above 
To answer some of the questions:
How do you apply to one?
You get in touch via the University/College directly regarding Application
What are the requirements?
Usually a level of knowledge of the language, depends on each College etc on what you need but you also need a certificate of sorts (like cert for completing the JLPT or similar) showing your competency in the language. Also just so you know a vast majority of college courses in Japan for foreigners are Post Graduate only. So you may need a degree before furthering education in Japan.
How much money is needed, etc
You're looking at anywhere from 10,0000 to 30,0000 (dollars or euros in general) per year. Thats just for some degrees i saw, postgraduate courses/masters could be higher. One particular course i saw was about 6000 euros ($7,500) per semester, each year had 3 semesters....
Also have to take into account the cost of living in Japan on top of that. It's an expensive country to live in.
Also bare in mind that Psychology degrees are useless on their own, you need a masters/PHD to get any sort of proper career prospects from it. So you will most likely have to dedicate the next 8-10 years studying. This is comming from two friends i personally know studying Psychology and on their way to also studying their PHD's.
Although i know it hasn't been an all positive post, i hope the information helped in some way. Keep searching for more answers though
