Best first-class train from Prague to Vienna, then from Vienna to Budapest

Look at Austrian State Railways - https://www.oebb.at/en/ and Czech Railways - https://www.cd.cz/en/default.htm for comparisons Going from Prague to Vienna some years ago , I used Czech Rail , which was fine . Vienna to Budapest , Austrian Railjet was my choice

On Prague to Vienna you will find both Czech and Austrian trains, and the CD and ÖBB sell tickets for both. So when booking you don't really know if you will be on a Austrian or Czech trainset. Booking on the Czech website does not mean a Czech train, and vv.
However the Czech an Austrian sets are near identical.

On Vienna to Budapest the Austrian sets are marked RJ, and the Hungarian EC. The RJ is better.

From what I read, the Austrian railway has the edge over Czech
railways for first-class, high-speed service on this route.

I'm not sure I agree. They run Vienna-Prague together and some trains are Austrian and some are Czech, and as mentioned there is not a huge difference between them. The really big difference is that the Austrian trains are red and the Czech ones are blue.

There is a way to check which companies train set you should be on through this website- https://www.vagonweb.cz/

Find your train number and it will show you the booked consist and the actual daily consist.

If I buy a first class ticket from Prague to Vienna but there are no reservations available for a peak travel day, will I be standing for 4 hours, or can I sit in the dining car?

If you have no reservation you have no guaranteed seat. If you have a first class ticket and end up standing you may request a partial refund of your ticket however.
So I would strongly suggest you get a reservation with your ticket.

If you cannot get your reservation at the time of purchase, because seat inventory has not been loaded yet you can always get the reservation later.