Spring is definitely a great time to visit. The weather is getting warmer, and the locals are usually in a good mood.
Temps in April and early May tend to be in the 60’s, and rise to the 70’s (and even 80’s by late May/early June). Spring showers are common, but typically they’re not all day affairs.
Personally, my strategy is fly as close to the May 15th off-season date as possible for a 20,000 mile outbound booking, and return in late May or June for 30,000 miles. That will save you a few miles, and insure mostly warmer weather.
I would suggest you look at weatherunderground.com. Enter a historical date for April and May, and click on “monthly”. You'll get a historical day-by-day weather calendar for the entire month.
As an aside, there is a terrific rail deal between Budapest and Vienna. A round-trip tickets cost about $35 if you complete your journey within four days. For a few dollars more, it includes free transportation on all of the subways, trams and busses for your destination city.