Not sure about Michigan even though I originally come from there. I would assume at least some of the bigger airports like Detroit DTW, Grand Rapids etc probably have some sort of back up generator.
Being that I live in Florida now and work at a large general aviation airport. We lost our power during Hurricane Irma. Pretty much after Hospitals and and 9/11 dispatch and other emergency services, airports come next on the priority list for repairs. The major airports have generators that would kick in so quick during an outage, the passengers there wouldn't even notice. At my airport particularly (which I am not going to name but it is in Central Florida somewhere) had generators to run the tower, lighting and fuel. The FBOs had their own generators. The only areas at the airport that lost power during Irma were our ops office, where we had a portable generator to run some things and some of the hangars and gates on one side of the field.
Not sure how I could answer for airports directly in the path of a Cat 4/5 Hurricane like some airports in the Panhandle after Michael that were likely leveled.
In the case of the natural disaster I am assuming that most commercial airports would run on generators (at least essential like tower, lights, fuel and terminals for passenger) and they will run as long as there is enough diesel to run the generators. Not sure if Michigan has the policy like Florida for putting airports at the top of restoring power after hospitals, water treatment, nursing homes etc