The last part of the journey is on a local train between Franzensfeste/Fortezza(I) and Innichen/San Candido(I) which sometimes continues to Lienz(A) and sometimes requires walking to a different train. ÖBB.at seems not to like interfacing with the Italian system to get a through ticket.
It seems to be possible to book one ticket from Flughafen Wien up to either Lienz or Franzensfeste. To Lienz the flexible price is €68.80 and it says that you can take any train on that day. That should also be the walk-up price, while I see fares as low as €25 for committing to a specific train, depending on how confident you feel that your flight will arrive on time.
As for the route, several options seem to be available some including Salzburg and/or Innsbruck, but I don't know how it works in Austria - as you were comparing to the UK, working out valid routes in the UK for a similar long-distance ticket would involve using the Routeing Guide, which comprises several tables that cover nearly 3000 pages if printed, and about 100 different maps. So maybe best to stick to the suggested route even if you have to take a later train (choose one on the ticket that runs reasonably often).
For the final train(s), Trenitalia seems to not want to sell online tickets at all. ÖBB.at will sell a ticket from Lienz to Bruneck for €14.70 but it says that is only valid for the train booked, so perhaps you should buy that at the station (or possibly online just before arriving there).