Looking for help with my timetable in Munich

Have you checked the on-time performance of the Euronight train? It averages 10 minutes late and has been as late as 2 1/2 hours in the last month. The train does stop for 1 3/4 hours in Salzburg, so that provides a buffer for delays between Budapest and there.

It does not stop at Munich Hbf, but only at Munich Ost, which is actually more convenient to get to the airport, and there are lockers at Munich Ost. You need to be at the airport 3 hours before your flight, so that means leaving Munich Ost around 10 am. I don't know what there is to do in Munich between 7 am and 10. The Viktulienmarkt opens at 8 am.

First, worth mentioning, you are taking on some risk by arriving the day of your flight. Budapest is a long way away, and given potential delays, mechanical problems, weather issues, labor strikes, if something happens, besides maybe a last minute flight, you don't have many other options to get to Munich. Now 90+% of the time, there are no issues, but I would at least draw up "Plan B", maybe consider changing your flight to the 5th.

But, to your question, if you arrive at 6AM, and need to be at the airport by 11 or 12 (depending if you have a direct flight to the US or a connection in the Schengen area), that gives you 5ish hours.

Your bags you can store at one of the larger stations, they will have lockers.

Maybe the biggest issue will be that at 6, nothing will be open, or at 7, some things at 8, some cafes and restaurants at 9, shops and stores maybe 9:30 or 10.

In early December, wandering the Englischer Garden, which would be a nice activity before a long flight, may not be as fun depending on weather. So I likely would head to the Marienplatz area, wander around, get some more pictures without the crowds. Some of the churches might be open early for at least entry and a look as well.

By 8 the stalls at the Viktualienmarkt will be opening, for something to eat or last minute souvenirs. At 9, you should be able to find someplace for more of a meal, my "go-to" place would be the Schneider Weisse Brauhaus for the classic Munich breakfast of Weisswurst, Pretzel, and a hefeweizen beer (they have other breakfast options as well). After that, you might be able to hit a shop or two, before finding your bags and getting to the airport.

I second the Schneiderweisse Brauhaus AND the Hefeweizen. I did that for lunch yesterday. The Munich Marathon was running past the front door. The oompa band was playing at the table next to us. It was all perfect. Just ignore the MacDonalds and pizza place across the street.