European East Pass

Booking well in advance Prague to Vienna and Vienna to Budapest can be as cheap as EUR 20 for each leg (have a look at the trustworthy Trainline.eu, which acts as an agent to rail companies at exactly the same prices). Those are non-refundable tickets for a specific train (day and hour). You will undoubtedly make reservations for your accommodation, which makes this hardly a disadvantage. Your intended side-trips will also be cheap. I would be amazed if some sort of rail pass would come cheaper.

Added:
Out of curiosity I looked up the ticket price for Prague to Cesky Krumlov. A single journey comes at CZK 199, which is EUR 7,70 or USD 9,00.

The European East Pass covers all trains operated by the national railways in Austria, Czech Republic, Hungary, and Slovakia. It starts at $249 per person for 5 travel days within a month in 2nd class. Extra seat reservations should not be necessary, including on Austria's faster "RailJet" service from Prague to Vienna and Vienna to Budapest

However, using our latest ticket-cost maps, I estimate that you'd pay under $200 per person to buy regular-fare, 2nd-class tickets as you go in each country for the destinations you listed. Prague to Vienna and Vienna to Budapest also offer advance-purchase ticket discounts, if you want to shave ticket costs down even more.

The Czech railway runs a direct 8:00 a.m. train from Prague to Český Krumlov and a return train at 14:00 that are covered by passes. But if your prefer the schedule and arrival location for a Regiojet bus, then that's a separate ticket, or if you take a connecting train, then the České Budějovice–Český Krumlov leg operated by GW Train Regio is not covered by passes.

I still like the concept of shuttle service, like CK or Bean Shuttle from Prague to Cesky K to Vienna. Cheap, efficient and nearly door to door. I checked next day fares a while back for Vienna to Budapest and even that was only 18 euro if my memory is correct. And you don't have to choose which train till the night before. Maybe Vienna to Bratislava to Budapest is similar. As for the Budapest to Eger train, cheap no matter when you buy it.

Thanks for each of your responses. Very helpful.