Innsbruck Airport: The sensible route is the airport bus/taxi to Innsbruck Hauptbahnhof, then an ÖBB Railjet/EuroCity/InterCity west to St. Anton am Arlberg. Allow roughly 1 hr 30 min–2 hr door to door and about €20–€40 depending on the rail fare; a private transfer is quicker only if your flight timing is poor, and costs far more. Do not treat the short airport-to-Hbf leg as the whole journey.
Zurich Airport: Take the train from Zürich Flughafen toward St. Anton am Arlberg—some services are direct, others require an easy change, commonly at Sargans. Allow about 2 hr 20 min–3 hr and roughly CHF40–70 bought close in; advance fares can be materially cheaper. The Arlberg Express shuttle is useful for door-to-door ski luggage or awkward arrivals, but at about €60+ it is not automatically better than the train.
Munich Airport: Use the S-Bahn into München Hbf, then rail via Innsbruck toward St. Anton. It is a long but workable rail arrival: roughly 4–5 hours total, typically €40–90 depending on how early you book. A private road transfer is only worth considering for a full group or very late flight; winter driving is not a casual shortcut.
Friedrichshafen Airport: Take local rail via the Lake Constance/Vorarlberg rail network, usually requiring connections around Lindau and/or Feldkirch, then continue to St. Anton. Budget about 3–4 hours and €30–60. This is a secondary airport option, not a painless resort transfer.
St. Anton am Arlberg railway station: This is the arrival point to aim for. It sits at Terminal West beside the pedestrian centre and the Galzig/Rendl lift side of town; Railjet, EuroCity, InterCity, Nightjet and WESTbahn services use the Arlberg line. Hotels around the centre are generally a short walk or local-bus ride away.