From Innsbruck Flughafen, take the F Line bus into Innsbruck to the main train station (about 20 minutes). Get your ticket (€2) from the driver. Taxis into Innsbruck should take no more than 10 minutes, and the fare is about €15.
From Switzerland to Vorarlberg in winter, there are buses from the Zurich airport Friday, Saturday, and Sunday several times each day for Zürs and Lech. You can book the transfer through the airline Swiss.