Thank you all for your help.
Free parking would be a problem, as it's a very central location.
The hotel's street is closed for traffic, you're only allowed to enter, drop passengers, and move on.
There's a parking garage, one minute walk from the hotel (at Mutter-ey-strasse). The price is, I think 1EUR/hour, I don't remember the daily maximum.
I stayed there this August, but only rented the car the day I checked out.
Get an answer from Hotel Alt Duesseldorf staff and past guests.