If you need to get an RP card, you need to submit for RP at the foreign police. You can go to foreign police only with mandatory reservation.
As there are many foreign students in September, the foreign police in Kosice is busy – as of today (6.9.2023) the next available slot in Kosice is on 18th of October 2023 (so in 5 weeks).
There is also option to submit at different police. The closest police with free appointments is in Michalovce, city 60 kilometers from Kosice. You can get appointment also in 2 days there, just you need to travel. As student, you have free train ticket
Please let me know which option do you prefer – a) submission in Kosice in 5 weeks OR b) submission in Michalovce on Wednesdays (next 13th, 20th etc)