It would really surprise me if this worked out well.
According to google maps, it takes at least 5.5h by car.
But this area (Köln/Düsseldorf) is AFAIK the most densely populated region in Germany, with the most traffic jams etc.
There is no guarantee that you actually make it in time.
http://www.wdr.de/verkehrslage/
It will be difficult to find somebody who has the time (Monday morning!) and willingness to drive you (not to mention finding somebody who actually shows up...) - and I certainly wouldn't drive it myself.
500 for rental is probably for a very cheap car. But doing 625km in a small car is no fun. Even less so under time-pressure.
I would take another flight from Düsseldorf. Or an earlier flight to Zurich, so you can take the train.