and then to set out to make trouble for Austria, with the aim to take her capital Vienna. Once you do that, Austria will automatically make peace.
It wasn't much of a mission reward for me because after I captured Vienna, the Austrians again declared war the next turn!

You want to keep a nice buffer against Prussia early in the campaign in order to deter them for as long as possible, while you deal with Schwarzenegger's fatherland.
It never works for me. I think the Prussians are hard coded to attack within the first 5 turns, so I end up doing a preemptive strike on them from turn 1.