Is the prayer that was performed at the university at 12:22, based on the observation that the Dhuhr athan was at 12:19, considered to have been performed before the legitimate time for prayer if the actual athan is raised approximately five minutes later, and must it be repeated?

The time for Dhuhr prayer begins when the sun declines from its zenith and extends until the shadow of everything becomes equal to its height. One can rely on the practice of trustworthy and knowledgeable udul (just individuals) concerning prayer times or on a reliable time calendar. If the calendar adopted by the university was prepared by trustworthy udul, then your prayer is valid and sufficient. There is no condition to hear the adhan, and the delay in the adhan you heard is not considered proof that your prayer was before the entry of its time, provided you relied on a reliable calendar.