What should a student do who heard some exam answers and wrote them down, and now regrets it and wishes to repent, and is repentance sufficient for admission to university, knowing that he fears he may have taken someone else's place?

If you intentionally cheated, you have sinned and repentance to Allah Almighty is obligatory upon you. Your repentance is sufficient to remove the sin. There is no blame upon you for applying to the college with your grades. Repent to Allah and do not repeat this act. As for what reached your hearing without any effort from you, there is no sin upon you for it.