Congrats on getting the job, it can be a lot of fun and there's likely great music! Primary duties will depend on the stage manager and how they like to divide up responsibilities on the team as well as on how many ASMs there are. ASMs are often responsible for Prop and/or Costume running documents as well as the WhoWhatWhere. The extent of that content is up to your SM. Sometimes it's just entrances and exits and applicable props/costumes, sometimes it includes tech cues as well. ASMs typically cue all entrances especially for chorus and supernumeraries (extras). Principals may take their own entrance cues, but you should be there in case they fall asleep. Each SM works a little differently and they should be able to guide you in this.
Rehearsal process: Often the first day is a musical sing through of the show, sometimes they just do music of the pertinent scene right before staging it. Actual staging rehearsal will look familiar as far as resetting props, having rehearsal costumes available. The conductor may request coaching time for a certain singer if he feels they need more work, that's usually private time and SMs are not needed.
Music reading fluency: I take it from the way you phrase the question that you have some familiarity with reading music. If so then the best way to become more fluent is to borrow a recording of the opera and listen to it while following the score. Then do it again. As often as you need. While one can often follow the vocal line that shouldn't be relied on in case the singer is off by a bar. Get used to following the orchestra (piano) line. If there's a particularly difficult passage you can either count beats diligently or even ask the accompanist for guidance in how to follow it, they are often helpful people.
Good questions. Let me know if you have more.
Beth Krynicki