Public transport: The most sustainable way to travel is by train (coming from Villach) to Hermagor. From Hermagor, you can continue by train shuttle, as well as bus / ski bus, to the town of Tröpolach.
Alternatively, there is also the option to travel by bus from the direction of Kötschach-Mauthen or Villach to Hermagor or Tröpolach.
From Tröpolach, you then take the Millennium Express up before reaching the starting point of the sledding track via a ride on the Sonnleitn lift.
Details at - Nassfeld mobility info and Kärntner Linien route planner
Arrival by car:
Via the B111 (Gailtal Road) to Tröpolach - from there via B90 (Nassfeldpass Road) to the junction right towards Sonnleitn (a few meters before the valley station of the Tröglbahn). Follow the road (L22a) for about 3 km (almost to the end of L22a) to the parking options.