i skied there a few days last winter. Flachau is the main place to ski from there. call in advance and book a FREE seat on the bus they run from the center of town to the resort. p/u is around 8a and the leave the Mtn around 330p
Rentals were not cheap and neither were the lift tix; so you may want to see if they offer discounted tix or rentals outside the ski area.
The area was a blast to ski.
I visited Salzburg about ten years ago for a week and skied five different excellent ski areas using the city's shuttle bus. See old trip report for info on some of the notable ski areas nearby (Kitzbuhel not discussed, but it is also within reach):
However, I think the bus fedegm referred to has stopped running for the season on Mar 16, but please verify, perhaps at this link: http://www.salzburg.info/en/sights/guided_tours/shuttle_service/salzburg_flachau_ski_shuttle
If you didn't want to rent a car it may be possible to take a train from Salzburg to Bad Gastein (~2 hours), but you would have to check on times. It has some higher elevation terrain that should guarantee April skiing. Perhaps some other ski areas are reachable via train as well?
Personally I would go for Flachau and Ski Amade.It's 70km south of Salzburg exactly next to A10, main highway connecting Salzburg and Villach. From Flachau you have whole Ski Amade (some 600km tracks) under your skis :)
Sure there are other options with about same distance (but not with highway connection, which means longer travel time), but even when it comes to skiing Ski Amade is pretty much best option, except for maybe Saalbach (90km away, but count for at least 1.5h drive if not more). Then of course you have Kitzbuehel, Kaprun, Schmittenhohe and several other smaller places.
Another thing is snow. This winter places north of Tauern didn't get much of snow, so Saalbach, Kitzbuehel..., might be in pretty bad conditions by 1st of April already, especially with temperatures we had in last few weeks (+15c at around 1000m high). Even with this Flachau and Ski amade got it a bit better, as even if they are still north of Tauern, they got much more snow then Saalbach or Kitzbuehel did.
PS: Fedegm there's no need to check for "discounted" lift passes, as there are none (ok there are few places where you can save 1eur). You might get discount price because 1st of April is way out of main season already, so for beginning and end of season, lift passes are cheaper, but you get this price anywhere, including cashier on bottom of lift. For ski rentals it's most likely true, you get cheaper price in some big sports store in Salzburg, then on little ski rental store on bottom of lift. Otherwise tickets are not so expensive compared to this what I heard about US. 40-45eur (around $55-60 I guess) for day ticket during main season. It's not little, but it's not end of the world either.