Budapest and Munich and Night train

Oooh, that’s a tough one. I don’t know about the night train, but there are people here who do.

It looks like you have 6 nights to allot. So maybe this?
May 1: arrive and disembark, sleep Budapest
May 2 & 3: Day in Budapest, sleep Budapest
May 4: day in Budapest, sleep on train
May 5 & 6: day in Munich, sleep in Munich
May 7: fly home

This is 3 mostly full days in Budapest and about 1 1/2 in Munich. If you decide to take the day train, I would do that on May 5.

2 days really isn't enough for Munich or Budapest and 4 days may be a bit long for Munich and a tad short for Budapest.

Still, you are in Budapest so I would take advantage of it and fly to Munich the morning prior to your flight, maybe hire a guide in Munich for a quick overview trip to help decide if you return for more on the next trip, then the best farewell to Europe dinner you can muster up and board your flight the next morning.

Lufthansa has at least 3 direct flights every day. $120 to $200 and 1:20 flight time.

But I am very, very biased.

That night train is 20:40 to 06:20.

So, checkout time is noon. Leave your bags at the hotel. And you are good to go sightseeing until maybe 5:30 pm, then a very early dinner and buy junk food and drinks for the 10 hour train ride. Then back to the hotel at 7:20pm for the bags, then be at Keleti Station by no later than 8pm cause you don't know the station and you don't want to get confused and miss your train. The train is about $80.

Wake at 5 am and now it's 7 am before you reach your hotel in Munich. It's too early to check in, so dump the bags and go sightseeing. Return to the room at check-in time for your first shower in 36 hours.

Or get up early in Budapest and head to the airport.

Your choice; but, although I love trains, with your timeframe I would probably take the flight over the day train, as well.

I love trains too. The Hungarian Railway Museum is perfect for that: https://www.google.com/maps/@47.5421866,19.097618,2a,75y,19.3h,95.49t/data=!3m6!1e1!3m4!1sjVl1jWv3O5Ao_B5qy6EY-Q!2e0!7i13312!8i6656?entry=ttu and https://vasuttortenetipark.hu/en/home/

Of course if you still prefer a plane:
https://aeropark.hu/en/museum/ and https://www.antique-airplanes.com/budapest-aviation-museum.html

Or if you prefer ground transportation:
https://kozlekedesimuzeum.hu/en

Note that, although this train can be booked on the Nightjet website it is in fact a Euronight train and is worked with modern MAV (Hungarian) sleeping cars, not Nightjet cars. The MAV cars may actually be better than Nightjet cars.
It actually runs to Stuttgart (beyond Munich). Test bookings indicate that it seems to be cheaper to book with MAV, rather than Nightjet.

The train pauses in the middle of the night at Salzburg to attach sections from Zagreb and Venice, and detach a section for Zurich.

Your section conveys one couchette car, one sleeping car and one seated car. There is also a seated section as far as Salzburg.

OR! You could pop down to Bucharest for 2 nights. Direct flights BUD TO OTP TO MUC are under $200 each. Several a day and about an hour to OTP and 2 back to MUC.

And, yes, more often than not best to buy your train tickets from the country of origin. The MAV site is pretty easy.

On when to be in Keleti:

For the dep. at 20:40 I would be get there at Keleti no later 19:30.

Arriving in Munich just before 6 AM is great, it'll be day light, food vendors are already opening up, and if you rather be in a station with none of the construction "stuff" to have a more leisurely coffee, breakfast, etc , I would suggest the U-Bahn / S-Bahn station, Karlsplatz/Stachus, a very nice and large underground passage.

This night train BP to Munich , either direction, is one I'll be possibly taking too in May, just a simple compartment seat.

Using Fred's advice, which isn't bad, dinner time is 5pm.