First of all, Austria definitely is a great place to ski at, and I think the best and largest area to ski at would be the Arlberg region (includes St. Anton, Lech and Zurs). It is unlikely you can ski all the lines at the Arlberg region in all 4 days, and while Lech and Zurs tend to be more expensive, look at the smaller villages in the Arlberg region like Warth/Schrocken or Zug where lodging prices might be cheaper. If you prefer convenience, St. Anton and St. Jakob are closer to the train station for easier access.

Ischgl is another great choice. Another great option is Solden along with Obergurgl/Hochgurgl, located in the Otztal valley. Solden is quite easy to access from Innsbruck and if you plan to have cheaper lodging, probably staying in Oetz might be cheaper and taking the bus daily to ski. Otherwise, stay at Innsbruck and take the ski bus to the various ski areas around Innsbruck if you like to try out different ski resorts each day. Finding lodging in Innsbruck should be easy but you might not get the ski chalet ambience.