What kind of crow is this?
I'd say you guessed correctly. This does appear to be the hooded crow (Corvus cornix). 


In German, it's called Nebelkrähe (meaning "mist crow").




 A Hooded Crow Corvus cornix in the garden of Belvedere (Vienna). [Source].  

Description: ashy grey bird with black head, throat, wings, tail, and thigh feathers, as well as a black bill, eyes, and feet.

C cornix is a Eurasian species that would be present in Austria:



Similar but incorrect species:

The only other similarly grey crow species I know are the grey crow (Corvus tristis), which is only present in New Guinea, and the western jackdaw Corvus monedula. The jackdaw is present in europe, but it can be ruled out because of its grey eyes and because of the grey feathers that creep superiorly from the birds nape onto its head (unlike your specimen). 


Note: I exhaustively checked all commonly recognized species of the Corvus genus