For students who are already in Japan, there is such a thing as domestic selection by university recommendation. But I guess the spots for that are relatively limited and quite competitive (lots of Chinese and Korean students already in Japan, self-financed but would like to have the scholarship).
Also note that in order to get *to* Japan and into your school, in order for the authorities to issue your visa, you are required provide some documentation (i.e. a bank statement) to show that you have enough funding available for at least your first year of expenses. A few years ago, the required amount was 1 million yen.
My advice would be to have a backup plan to go to school in the US for at least your first year, and apply next year if you absolutely want to do the full program at Sophia.
It probably would also be wise to keep your mind open to doing your undergrad in the US.. it's MUCH easier to get in on a one-year exchange program than to get the monbukagakusho scholarship for undergrad, and you might find one year to be quite enough (I certainly did at the time).