Hi Valerie,
Well, beeing Swiss I can't tell you much about Austria but about the nearby swiss valleys. I guess the avifauna is quite the same.
The Inn valley (from Landeck up to the swiss engadin) is a very dry valley. It holds good numbers of Whinchats, Red-backed Shrikes and Corncrakes. Look out on the sunny slopes, facing south. Also Barred Warbler can be found here, at the edge of its distribution.
If you have time and a car, drive down to Landeck, then up to Switzerland and to the Swiss National Park. Golden Eagle is common and Bearded Vulture (or Lammergeier) can be seen, although most of them breed in captivity and are released as young birds. All typical Mountain Species can be found.
In Ischgl, go up as far as you can with a cable car or something. Snow Finches like to feed near mountain restaurants. Also Alpine Accentor can be found sometimes easily around the mountain resorts.
If you go hiking, look out at the treeline, the best habitat for Citril Finch. They usually sit on the top, but are still hard to spot. Other species like Ring Ouzel are also most easily seen around the treeline. Like in the lowlands: The early bird catches the worm or the early birder sees the bird. Go there, if possible, very (in my experience everything between 4.30 am and 6.00 am is fine) early in the morning. You will be rewarded.
Usually, in the mountains, there are no special spots for the birds, they can be found virtually anywhere (expect where there are to many people).
If you walk, sit down (also here, around the treeline is very good or in rocky areas) and wait. Suddenly birds (Wheaters, Pipits etc.) will pop up. If you are lower down in forests, make a break when you find a more open area with a overview and just wait and see.
Telescopes can be very rewarding in the Alps but they are hard work to carry around. So this decision is up to you and to your fitness. Generally, especially if you're not used to mountains, birding downhill is much easier then uphill, at least you can hold your binoculars quitly.
If you want to try for owls: Best time after 11p.m. in the middle of some dark, mature forest.
Good Luck and enjoy your holiday!
André