Customs at Munich airport

To what destination?

Given the vagaries of border control processing times and the current issues with baggage processing, stacked on top of 'will the flight arrive on schedule', buying once you are there may be the best option.

It really depends on the terminal and how many other flights are arriving at the same time. Typically I've gotten off the plane, through customs and on an S-Bahn in just under an hour but I've had to wait up to 2 hours before.
There should be no need to book any train tickets before getting to Munich as every train leaving the airport is an S-Bahn or regional train that requires no advance purchase.

DJ

As long as you have nothing in your bags to declare at customs, it won't take any time. After you pick up your bags, you'll just walk out through the line that says "nothing to declare", and you are out the door. What could take a little time is Immigration (incoming passport check), if Munich is your point of entry into Schengen. However, that has never (knock on wood) taken me a significant amount of time, even coming off a jumbo jet with lots of people.

I've never arrived at T1; always T2. Immigration is at the center of the concourse on the same floor as out arrival. (This could be different today because of the new satellite T2.) After clearing immigration you go down elevators to the baggage retrieval room and customs.

Because I do carry-on, I've never had to wait for my bags to come to the retrieval room. Last time I arrived in Munich, as I walked out through "customs", I looked back to see the young lady who was sitting next to me on the flight over, still waiting for her luggage.

If you are going a significant distance from MUC to your destination, you might be tempted to book a long distance train with one of those advance purchase, Savings Fares. Don't. If your flight arrives too late to make the connection, you lose the fare. Instead of leaving a time gap in MUC for an express train, took at the time difference using regional trains and a Bayern-Ticket. You might find that you can get to your destination just as quickly by taking the next regional train rather than waiting for a long distance train you think you can catch.

Useful comments all, thanks. This will be our first stop from US, so will need to go through passport control. Then train to Innsbruck. Sounds like purchasing tickets upon arrival is the best idea. I presume there are ticket vending machines similar to others I’ve used in Europe.