In short, you will need to be an amazing prospect. If we are talking grades, try 4.0 or 3.99. You should have passed the highest level of the Japanese Proficency Test, which means you have the basic fluency of an average high school graduate. Since it is a technical field, you will need to know all the basic vocab for your area, or, you will have to be much better at Japanese in all areas (reading, speaking etc.) than an average high school student. You should be at the level of having a realistic shot of getting into John Hopkins or the like if you were going to go stateside. Again, you will need to sit the schools entrance exam, which are usually only given once a year and in Japan, in Japanese. Your Japanese will need to be perfect. What Tiger was trying to point out, though a bit roughtly, is that true fluency, idioms, expressions, absolute perfection needed to be a doctor takes years of living in Japan. Books will only get you so far. I think Tiger was trying to point out that you might be a little schort on time if you want to be in school before you are 26, 27 or even 30. You want to be a doctor, and doctors hold a special place in most all societies. You will need to be perfect.
Given the level of xenophobia in Japan just about common issues, I can imagine the problems many people would have visiting a foreign doctor for mental problems. On top of that, you are dealing in a field most Japanese people won't use due to the stigma attached to it. You have to first overcome your foeign-ness, which some people will never get past no matter how brilliant you may be, then overcome the social stigma of psycologic disorders. IMHO the best shot you have is to go to school in the US, get some years under your belt, then start looking at the more international oriented hospitals in big cities. They will get you set up on your visa and facilitate your getting through governmental loggerheads. Additionally, hospitals might take more favorably to your application if you train in the US and spend a year or so on exchange in Japan. I don't know of any such programs off-hand, but most the top-notch doctors in Japan have spent time in the US. Also, if it works for medicine like it does in some other fields, I think if you ever want to go back to the states and still work as a doctor, you will find it much easier to find a job with a degree from a US institution than a Japanese one, unless of course you go to some wonderful Japanese school. Then after all that, if you really love it here, you can look into the nightmare that is setting up a your own business in Japan (once again, your foreign-ness will work against you, unless you become a Japanese citizen first).
Is it possible? Yes, anything is possible if you work hard enough. You wanted to know what you might be up against and I took a crack at it. I'm not trying to break down your dream. But success will take an amazing, amazing effort from you, a good deal of time, and a good deal of luck along the way.