Itinerary Central Europe

Good plan, but I suggest visiting Salzburg after Vienna.
Also, consider a day trip from Salzburg to Berchtesgaden.

You don’t need a car and for a trip like this it would probably be more trouble than help. Trains and CK Shuttle are fine for all of it.

I moved Salzburg and Vienna around as suggested above.

I removed some nights. Not that there was anything wrong with what you had, but for me (and we are all different) the night I took out of Salzburg and the night I took out of Prague are “extra nights”. Put them back after you consider all the other options.

• Another night in Budapest or Vienna
• or a night in Gyor on the way to Vienna and see the archabbey at Pannonhalma,
• a day trip from Budapest to Szentendre or Tihany
• or a day trip from Vienna to Bratislava or Melk,
• or put the days back and do a day trip out of Prague to Kunta Hora
• or from Salzburg to Hallstatt.
  

4 nights Budapest

• Train to Vienna

4 nights Vienna

• Train to Salzburg

2 nights Salzburg

• CK Shuttle to Cesky Krumlov

2 nights Cesky Krumlov

• CK Shuttle to Prague

3 night Prague

• Flight home

Hotels
Budapest, you can get a darn nice hotel at a great price, so that’s my recommendation: https://community.ricksteves.com/travel-forum/hungary/budapest-updated-hotel-list
Vienna, they tend to be a bit more pricy which is why Vienna and Budapest balance so well. Just get as close to the Ringstrass as you can.
Salzburg. I have only been once and I stayed at the Hotel Elefant thought is pretty perfect for my trip. I would do it again.
Cesky Krumlov. I stayed at the Hotel Dvořák which isn’t getting the greatest reviews these days. Shame because the location is perfect.
Prague, the Hotel Ventana, best location in town and one of the nicest hotels I have ever stayed in.

Can you add a night in Vienna so you can take a day trip to Bratislava? It only takes an hour by direct train: https://shop.oebbtickets.at/en/ticket. You’ll need to catch the train at Wien Hbf (Vienna’s central station) to Bratislava hl.st.
After you arrive at Bratislava's train station, take Uber to Michael's Gate (Michalská brána).

Hey!
I am also doing a West to Central Europe trip in April this year.
I also have Czechia and Austria in my itinerary and based on all the research I have done it will make more send to go to Vienna first from Budapest and then to Salzburg.
Since I am coming from West this is how it looks for me:
Prague ( 4 days) -> Cesky Krumlov (1 Day) -> Salzburg (3 Days) -> Vienna (3 Days)

I have booked a bus which runs directly from Cesky Krumlov to Salzburg which will take 3 hours 15 minutes so it is convenient as well.

Please let me know if you have any queries which I can help with.

Cheers and Happy Travels!