I have just completed a similar itinerary. I flew into Budapest on 19 April; went on a 7-day long-boat cruise down the Danube; returned to Budapest, where I hired a car to take me to Bratislava for lunch in the afternoon en route to Vienna, where I stayed for 3 nights. Then I hired another car to take me to Cesky Krumlov, where I stayed over-night at Pension Anna (mentioned in Rick's tour guide). I then took an easy bus ride to Prague, where I stayed 3 nights. I then flew to Paris, where I've been for 12 days. I'm en route to Amsterdam tomorrow for 3 nights. Then home to Florida in USA.
While I cannot help you much with your question about the economy of eurail vs separate tickets (although I'm thinking individual tickets and a bus ticket from Cesky Krumlov to Prague might be cheaper), I can tell you that, if you keep moving, you can probably do it all in 12 days! Depending upon what you want to see and do in each city, a minimum 2 or 3-night stay in each Budapest, Prague, Vienna, and Berlin would be ideal. Bratislava and Cesky Krumlov can each be visited in an afternoon--follow Rick's walking tours, which you can do in 2-3 hours including lunch!
To make the most of your time in each place, again, follow Rick's walking tours and enjoy Big Bus, Hop-on-Hop-off tours in the big cities like Budapest and Vienna. (Be sure to book the City by Night tours as well since these cities are magnificent to see at night!) In Prague, I booked a wonderful walking tour with a company called Tours for Charity (proceeds go to local charities helping needy children). It meets at the Municipal House daily at 10:00 am.
I hope you this helpful. Have a wonderful trip!