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I can't speak for Austria on a whole but there are things you can do in Vienna on the cheap or Free.

Viennese Museums
The cultural variety in Vienna can be overwhelming and might unnecessarily stress your budget. High costs are avoidable if you know which museum is free on which date. Certain museums like:
Geldmuseum
Kunsthalle am Karlsplatz
Bezirksmuseen
MUSA
…are always free for everybody.

Children and teenagers (up to the age of 19) enjoy free entrance to the following museums:

Albertina
MAK
mumok
Belvedere
Technisches Museum
Naturhistorisches Museum
Kunsthistorisches Museum
Lichtensteinmuseum

There are free entrances for everybody on the first Sunday of every month to the following museums:

Uhrenmuseum
Wienmuseum Hermesvilla
Haydnhaus
Wien Museum am Karlsplatz
Heeresgeschichtliches Museum
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Otto Wagner Pavillion

Summer Cinema at the Rathausplatz
There are events in front of the Viennese Civil hall all year round. Most of them are free of charge. A special highlight is the summer cinema which takes place at the end of June until the beginning of September.
At nightfall movies from the world of jazz, opera, ballet and music from around the world are displayed on a 300 square-meter screen.

Tip: Seats can’t be reserved, that’s why: first come, first served applies.

Free concerts
The „Donauinselfest“ is a free open air music festival with a long tradition. Every year since 1894, this three-day long party is visited by around 3 million people.
With 11 stages and over 600 hours of program it is the biggest open air festival in Europe.

Tip: You will find more Infos here:

Stroll the gardens of Schönbrunn
The baroque castle Schönbrunn not only is a world heritage site, but also the most visited sight in Vienna. The habsburg castle is almost entirely in it’s original condition. You can stroll around in the beautiful garden, that used to be for the imperator’s family only.
Many runners use the gardens for their morning workouts. You will get a special view over Vienna and the castle Schönbrunn if you walk up to the Gloriette.

Saying that cities in Austria have City Cards that are valid for 24, 48 & 72 hours. They include free entry or reduced entry into places and free public transport. & Other places offer similar things so it's worth checking with the local tourist board.