For the trip from Koblenz to Kronberg you should use a taxi company from Koblenz. I found this site Taxi Diekel Koblenz. They offer airport transfer to FRA for 135€. Since the Schlosshotel is about the same distance the price is probably the same. Anyway only way to fnd out is to give them a call and tell them what you want and how many people you are.

An alternative is to take the train back to Frankfurt and then take a cab from there to the Schlosshotel. At the large train stations are enough taxis waiting.

For the transfer from the Schlosshotel to the airport the hotel will call the local taxi company for you.