Like everyone else here who hasn't got the results yet, I am waiting on tenterhooks to find out if I got the scholarship or not:nailbiting:. I thought it might be worth posting my experience though because I did not apply via embassy, but via my university. Weirdly enough I haven't been able to find anyone on this forum or elsewhere who applied through their uni, so I'm not at all sure what my chances are or how the final decision might come about, but the process so far has been pretty similar to the embassy route.
The first screening that corresponds to the language exam at the embassy is sending your application to the head of your subject area/whoever is in charge of Japanese studies at your university. Passing the screening requires a good application and, IME, making a good impression in general on the faculty members in charge. Obviously, there is no additional language exam as your university already knows your level and I was told that the teachers who knew me were also consulted for the final decision. My university could nominate only one student and fortunately I was the one they chose. They forwarded my application documents to my target university along with their recommendation letter before the deadline in February.
In the middle of March, I was contacted by the university I applied to for an e-mail interview (going by what students applying for the grad scholarship have written, it is possible that some universities will request a Skype interview instead). I was asked about what sort of study I'd like to do in Japan, and how studying abroad would help me in the future. Then we chatted a bit more informally about my studies so far and my reasons for studying Japanese in the first place, why I chose to attend my university, my study/employment history etc. It was pretty nerve-racking but I think I was able to handle it well. I also asked her when the results would be announced, and she said June at the earliest.
I didn't actually know whether I'd passed the second screening (university interview) until a week ago when I e-mailed the International Office at my university wrt my exchange student status, as I had to make a separate application for exchange studies (so my Japanese credits would get transferred back to my home uni, basically). This was accepted in March but I hadn't heard back from the Japanese uni yet, so there was a bit of e-mailing back and forth, and I found out that they had forwarded my MEXT application to the government and were hoping I would get it! So, I have passed the second screening as well. I hope that since the Japanese university I applied to has accepted my nomination, the government will approve the scholarship too.
Sorry for the essay, but it might be helpful to anyone who is also applying via their university. I'll definitely post an update once I know the final decision, and if anyone has any questions, feel free to ask!