Looking at the challenge from a different angle, could the ASM call the show while I run backstage or simply become another crew member that makes sure everything gets done safely? My reasoning is that the ASM has sat in all the rehearsals and knows the show well, and is accustomed to the original stage manager's style. Not only that, but the ASM has just been through tech and is intimately acquainted with the show. The ASM would also know the actors' habits and blocking, such as how long an actor's pause will be inbetween lines and when they're going to cross to the other side of the stage.
In this scenario, since the props have already been preset I would introduce myself to the cast and crew. I would then go over the run sheets that the ASM would hand-off and notate both what the ASM does during the show and particularly difficult areas. I would be asked to get a quick tour of the theatre (if I don't know it already, assuming I go to the same college), mentally taking down where the green room, dressing rooms, and booth are. Double-check the safety kits as well as know what is in them. I would then re-check the props to familiarize myself with the stage and the show. In addition, I would carefully go over the QC and request whether the Wardrobe Supervisor (if there is one) could be on deck that night. If not, then I would ask the crew members who do the wardrobe changes to give me a quick talk-through of what happens and how long exactly each QC has.
However, in the situation put down by PSMKay, I agree with Thespi and BLee methods, and would add introductions and updates to the cast, crew, house management, and teachers/supervisors (whoever is overseeing the show - the latter can be done through phone or email if necessary). I understand that in certain situations it is more difficult to replace the ASM with a new person than it is for them to call the show, especially if the ASM has an extremely busy run sheet.