I am 24 full years old There is a 'certificate instead of a military ticket' about...

At the university, you were told correctly, because Certificates instead of a military ticket, which were previously issued by commissariats, are not military registration documents and must be replaced.

"is it mandatory to pass the medical board to get a prescription in this case?" - YES, definitely. In accordance with Article 14 of the Law of Ukraine "On Military Duty and Military Service", in order to obtain a certificate of registration, it is necessary to undergo a medical examination.

"are there any nuances that will lead to the fact that I will be forced to abandon my studies in favor of serving in the army?" - no one will be able to force you to abandon your studies. However, if the VLK recognizes you as suitable for military service and at that time you will not have a formal deferral, in accordance with Article 23 of the Law of Ukraine "On Mobilization Training and Mobilization", then you may be issued a summons for dispatch (mobilization order). Failure to appear on such a summons entails criminal liability. In addition, if you will be enrolled in the ranks of the Armed Forces of Ukraine, that is, if you will acquire the status of a military serviceman, then you will not be able to leave the military service on the basis that you are in graduate school.

"what actions should be avoided when passing the medical examination in my case, if I want to continue my studies?", - in order for you to continue your studies, it is necessary for the VLK to recognize you as unfit for military service. In the event that the VLK recognizes you as suitable (or limitedly suitable) for military service and you do not have a formal deferral, then you may be called up for service and then it will be quite problematic to be released from it.