What should one pray if they slept through Asr prayer and woke up five minutes before Maghrib Adhan; Asr or Maghrib first, and what is the ruling on the order of missed prayers?

A Muslim must perform the 'Asr prayer before its time expires, i.e., before sunset. It is not permissible to delay it until the time for Maghrib begins. If one manages to perform one unit (rak'ah) of 'Asr before sunset, then they have caught the prayer.

If the sun sets before the 'Asr prayer is performed, the issue becomes one of sequencing between the current prayer and the missed one. The majority of jurists (Hanafis, Malikis, and Hanbalis) hold that sequencing is obligatory between missed prayers themselves and between them and the current prayer. One should first make up the missed prayers and then perform the current prayer, unless the time is so short that it only suffices for the current prayer, in which case the current prayer is prioritized. As for the Shafi'is, they believe that sequencing is a Sunnah (recommended) and not obligatory. It is recommended to prioritize missed prayers over the current one unless there is a fear of missing the current prayer. Sequencing is waived according to the Hanafis and Hanbalis in cases of forgetfulness or fear of missing the current prayer. The Hanafis further add that sequencing is waived if the number of missed prayers exceeds five.