Like in many other countries in Europe, most restaurants are closed between lunch and dinner time. Thus, you will usually find warm meals between 11:30 - 14:30 and 18:00 - 21:00. 

Of course in large cities like Innsbruck and Salzburg you will find some exceptions like breweries or bar restaurants who serve warm meals all day long.  

As a greek when I visit Innsbruck, I am usually dining at Restaurant Posidonas, which is all day long and till midnight open. Of course if you want to eat fast food, you will have a lot of options.

Mittagsmenüs!

If you want to eat lunch in large cities, search for Restaurants who have Mittagsmenüs. These menus have discounted prices to attract business people, and of course you can also benefit from it.