The website www.gastein.com provides a good guide to the Gastein valley and the attractions it offers all year, but especially Winter and Summer.
As the apartment has the option of self-catering, it is useful to know that Bad Gastein has a large and well stocked EUROSPAR, located near the Bahnof. Most daily supplies can be sourced here. A visit to the Eurospar from the Schillerhof will require the use of transport as it is a bit far to carry supplies back.
There are many bars, restaurants, coffee shops etc. in the town that are 5/10 minutes walk from the Schillerhof. Notable après-ski favourites are The Silver Bullet and the Eden Pub, but there are many others all over town.
There are ski and equipment hire shops in Bad Gastein, these are some suggestions:
http://www.ullreich-gastein.com/
http://www.intersportrent.at/skirental/shops/Bad-Gastein/16/
http://bad-gastein.skiset.com/ski-rental.html
http://www.snowrental.net/skiing/austria/bad-gastein/ski-hire-bad-gastein.html
Both Intersportrent and Hervis are close to the main Stubnerkogel lift.
In the summer months, Intersportrent does cycle hire.