Carrying a firearm and the ammunition intended for use with it from EU countries and third countries is subject to various restrictions and requires a permit under certain circumstances.
To carry a firearm and the ammunition intended for use with it from EU countries, Switzerland or Liechtenstein into Austria or vice versa, you will require a European Firearms Pass in which the firearm is entered, plus a permit to carry that particular weapon. Hunters and sports shooters do not require a permit; instead, they only require the European Firearms Pass, provided that they are able to prove that a particular hunting or shooting activity is the reason for their trip. This does not, however, apply to handguns carried by hunters.
Travellers domiciled outside Austria must carry a permit for the possession of firearms that require official authorisation pursuant to § 39 of the Austrian Weapons Act. This permit can be obtained from the Austrian representative authorities abroad. For category C and D firearms (which includes the majority of hunting weapons!), there are no import or transit restrictions from non-EU countries (except for Switzerland and Liechtenstein).
Please note that there is a general ban on the import of certain weapons (e.g. pump-action shotguns or knuckledusters).