My admission to the University (Use 'the' if you had already named the University)

In that year (Not wrong but use this if you had previously referred to a specific year) This year, I finished studying at school but I must  had to have passed my exams. They were very difficult.
When the exams was   had ended, I collected my (maybe 'the' depending if they were yours or not) documents and gave their  them to the University. This University was Siberian State University of Railway Transport (SSURT). But I was very sad, because I wanted admission to the Ryazan Higher Airborne Command School (RHACS).
I know I will study there. Maybe, next year RHACS will do women's conscription. I am awaiting this moment! (Also you can say, 'I am waiting for this moment')

Well done!! ^_^

My admission to the University

This year, I finished school. But I had to pass my exams. They were very difficult.

When exams were ended, I collected documents and gave them to the University. This University - Siberian State University of Railway Transport (SSURT). But I was very sad, because I wanted admission to the Ryazan Higher Airborne Command School (RHACS).

I know I will study there. Maybe next year RHACS will do women's conscription. I am waiting for this moment!

Good luck with your English, and with RHACS!