The parking area in front of Schönbrunn palace charges €3.90 per hour, up to a maximum of €39, i.e. after the 10th hour you won't be charged anymore. By the way, this is a normal price for a commercial parkig lot or garage.
You are not intending to stay overnight at the palace? :-)
In Vienna there is short-term parking everywhere without being designated by road signs. That means parking is limited to 2 hours (except Saturdays and Sundays) and you have to pay for it by buyig parking tickets in advance at a tobacco shop (Trafik) or via the app "Handy Parken". See here: https://www.wien.gv.at/english/transportation/parking/shortterm.htm
There is no way of parking your car on the street for a longer period, unless you are a (also paying) local of that particular city borough. Exploring Vienna by car is the worst idea possible. Compared to publig transport you will lose a lot of time, and have the hassles of parking. Garages are nearly everywhere, but you will be charged accordingly.
This is a website (in German only) for searching parking lots or garages: https://parkplatzsuche.at/kurzparken/Wien
The cheapest garages are the P+R Garages (park and ride), always located at major public transportation hubs. They are subsidized and charge about €4 per day. See here for a list: https://www.wien.info/en/travel-info/to-and-around/park-ride-354906