Is it permissible for me to use nepotism to gain admission to university, knowing that nepotism is widespread and I may be more deserving of acceptance than others?

There is no impediment to your friend interceding for you at this university, so long as you are eligible for admission and have fulfilled its conditions, provided that you do not take the place of someone who applied before you and has a greater right to admission. The other student's obtaining admission through illicit means does not make it permissible for anyone else.