Public transport: The most sustainable way to get there is by train (coming from Villach) to Hermagor. From Hermagor you can then reach the town of Tröpolach by the station shuttle and the bus.
Alternatively, there is also the option to arrive by bus from the direction of Kötschach-Mauthen or Villach to Hermagor or Tröpolach.
From Tröpolach you take the Millennium Express up to the mountain station. Here you change to the Madritschenbahn and ride it down to the valley station - now just a few minutes to the border.
Details at - Nassfeld mobility info and Kärntner Linien route planner
Arrival by car:
Via the B111 (Gailtal road) to Tröpolach - from there via the B90 (Nassfeldpass road) to just before the Austrian/Italian state border.