Is it permissible for a university student living in Cairo to go on a trip with the university to Wadi El Rayan, where they depart on Friday morning and return on the evening of the same day, without a mahram, due to the absence of a brother and the difficulty of convincing the father?

It is not permissible for a woman to travel for leisure, tourism, or similar purposes without a mahram (definition). We do not know of any scholar who has permitted this. Paradise is surrounded by hardships, and the Fire is surrounded by desires. Adhering to the truth in times of corruption and scarcity of helpers brings a greater reward.