My general solution to this problem is to strike preemptively. As Stage Manager for my collegiate productions, I generally ask my ASM(s) to arrive 30 minutes prior to rehearsal start times (of course, depending on the set-up procedures). In addition to that, I arrive at least 15 minutes prior to that. Since my travels to rehearsal are generally only across campus with no driving involved, I am only detained by perhaps a class running over, or weather conditions (I go to school in Rochester, NY: the SNOW BELT). The 15 minutes allots this buffer, but in the event that I am on-time for my personal call, I have time to 'zen out,' and have a built-in break in my otherwise hectic day. I look forward to this time every day. Often, it is my time to have a 'relaxed' dinner, at least compared to my lunches which generally occur as I walk between classes.
Similarly, I make sure that I call myself at least 30 minutes, if not 60 minutes, prior to the earliest call for actors or crew during a performance run. Again, I give myself a buffer of 'Oh crap, my car won't start' or traffic or weather or whatever, but I also give myself time to relax and get myself into the right frame of mind for the show.
As to the specific challenge of running late for a performance in an emergency-type situation, I would definitely call my ASM(s) to instruct them in what was needed to check/set-up/call/etc. Hopefully, I would be able to give them a time estimate for my arrival, and hopefully, I make it in time for the scheduled curtain time.
However, this has provided me a new insight (learn something new every day, huh?). I have never installed a 'back-up' system in my chain of command, such as training an ASM to cover the prompt book and a crew member to take over for the ASM. I am going to incorporate this into my upcoming projects.