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Mariä Himmelfahrt is a local holiday in Saarland and in the predominantly catholic communities in Bavaria (1.704 out of 2.056 communities). The further south you are in Bavaria (such as in Munich), the higher the likelyhood of it being a holiday.
Employees may not be employed on Sundays and public holidays from midnight to midnight. (§9(1) (ArbZG)).
§10(1) (ArbZG) contains (a German only) list of professions where exceptions can be made.
A summery of all of these rules, specific to Munich, can be found here:
Verkauf von Reisebedarf (Zeitungen, Zeitschriften, Straßenkarten, Stadtpläne, Reiselektüre, Filme, Tonträger, Bedarf für Reiseapotheke, Reiseandenken und Spielzeug geringen Wertes, Lebens- und Genussmittel in kleineren Mengen sowie ausl. Geldsorten)
Sale of travel supplies (newspapers, magazines, road maps, city plans, travel literature, films, sound carriers, travel first aid kits, travel souvenirs and toys of low value, food and luxury goods in small quantities as well as foreign types of money)
Generally speaking, shops that deal with travellers in places such as train stations will be open. The further away from a travel hub you are, the less likely it is to find shops open. Shops that state that they are offering some form of Reisebedarf (travel supplies) are generally allowed to stay open when others must close.
Restaurants, cinemas, zoos etc. are generally open.
Each museum will decide for themselfs if they open or not. If you see something like Feiertags geschlossen in their opening hours, then they are closed.
From @mlk: Gemeinden in Bayern, in denen „Mariä Himmelfahrt“ 2021 ein Feiertag ist is an official map [as PDF] which marks all the communities where it is a holiday in blue.
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I don't know Munich personally but here's a broad strokes answer which should apply to large German cities (you should expect significantly fewer options in smaller cities, towns or villages).
Almost all shops will be closed, due to legal restrictions. There are two main exceptions:
Most restaurants follow their usual hours on holidays (other than Christmas, New Year, and perhaps to a lesser extent Easter). Some may be shut (in fact, it's not unusual for restaurants to have Mondays as a Ruhetag - day off - anyway) but you shouldn't have much trouble finding somewhere to eat.
The same largely applies to museums. They typically follow their usual hours on holidays, with the same exceptions. It's also not unusual (but far from universal!) for them to be shut on Mondays even when it's not a holiday. There's no general answer, you should check the website for whichever museum(s) you're interested in. Most have English versions available, which may be somewhat limited but should at least contain opening hours and holiday information.