It would be easier, you would have less ground travel time and I doubt it would cost any more if you would fly into Prague and out of Budapest. But using in and out of Vienna, this is possible:
Saturday, 22 April, leave US
23 April, Arrive Vienna
24 April, Vienna
25 April, Vienna
25 April, Train Budapest (about 3 hours)
26 April, Budapest
27 April, Budapest
28 April, Budapest
Saturday, 29 April, Morning Train to Salzburg (about 5 hours) then figure out how to get to Hallstatt
30 April, Hallstatt
1 May, Hallstatt Cesky Krumlov by Bean or CK Shuttle (about 3 hours)
2 May, Cesky Krumlov to Prague by Bean or CK Shuttle (about 3 hours)
3 May, Prague
4 May, Prague
5 May, Prague to Vienna by Train (about 4 Hours)
6 May, Vienna
Sunday, 7 May, Fly Home