Is an 80-minute stopover enough time to clear Immigration at the Munich Airport?

Cusatoms and immigration in EU/Schengen are seperate.

Immigration is handled at your point of entry to the Schengen area. In your case that is munich. Customs on the other hand is normally handled at your final airport, so you won't have to collect and re-check your baggage.

If the airline is prepared to sell you the flights on one ticket that means the airline thinks you will most likely make the connection. If you buy the flights on one ticket and are delayed on your incoming flight or held up in immigration then it's the airlines job to sort you out a replacement connection.

OTOH if you buy the flights on seperate tickets and you get delayed you are going to have to fork out for a new flight.

Assuming you have the trip on one ticket:

If the airline sold you such a ticket, there is enough time - otherwise they wouldn't sell it.
Anyway, an hour is easy to do in Munich. Worst case, if you miss your connection, you get a free rebooking on the next plane - and there are many to Vienna.

Your bags are checked through; german customs doesn't care about them.

Outside of the USA, basically all airports are 'international', so only the final arrival needs customs to inspect your bags. Only in the USA exists the concept of 'non-international' airports, so you are forced to immigrate and pass customs on the first ('international') arrival - your connection could be to Cornfield, IO, which might not have customs and immigration.