Leaving the highway, Klagenfurt Airport can be reached easily because of its own exit on the motorway.

From Graz:
Motorway A2 direction Italy - leave the Motorway at the Exit Klagenfurt Flughafen and follow the road signs. The Airport is roughly 1km from the Autobahn exit.

From Villach:
Motorway A2 direction Vienna - leave the Motorway at the Exit Klagenfurt Flughafen, and follow the road signs. The airport is located about 600m away from the exit. (Caution: the exit lane to the Exit Klagenfurt begins in the Ehrenthalerberg Tunnel, be sure not to miss it