I'm planning to go to Salzburg (Austria) by plane, to ski.
How are the connections between the ski runs and the airport or the city?
Yes. It is easy and you have several options. From the Wolfgang Amadeus Mozart Airport, you can easily reach ski resorts located in the Salzburger Land or even in Tyrol. There are buses. You also have the possibility to take a taxi or shuttle. Eventually, you can rent a car.
As an example, Faistenau and the Fuschlsee are only a 30 minutes car drive from Salzburg. This means that you can easily reach it in an hour by bus. The big resorts, such as Kaprun / Zell am See or the Skicircus can be reached by a shuttle in roughly 90 minutes from Salzburg airport.
If you intend to stay only a couple of days, like an extended weekend, Faistenau can be a good choice. It's cozy and the ski resort is big enough. And it is not too far from the airport.
If you want to check distances and travel time, you can use Google Maps. That will give you a good estimate. Details for buses and shuttles are available via the airport's website, by using the links given above.