I don't mean to use a tent, just a mat + a sleeping bag, or possibly a bivy sack if the weather is foul.
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Yes it is, but only for one night in any single spot. Staying multiple nights is considered camping and is not allowed.
It is never allowed in national parks.
I could only find Dutch references for this, but they all say the same and refer to relevant text in the Austrian lawbooks.
What you describe is absolutely allowed (using a bivouac to spend one night).
If you set up a small camp (described as a "planned bivouac" further in this text), you shouldn't be in a protected area.
Sleeping one night without a tent or a small igloo.
An emergency bivouac is basically allowed everywhere. A planned bivouac is normally tolerated as long as it isn't set up in a protected area (official German term is "Schutzgebiet"). It's explicitly prohibited in such regions (ask the administration of these areas for further information).
Spending one or more nights in a small tent apart from official camping sites.
The detailed regulations differ from state to state in Austria.
You should use a safe camping stove to cook. Open fire is prohibited in protected areas (official German term "Schutzgebiete") and in the forest.
Detailed regulations are very different from region to region.