No problem at all to get a taxi.

Though, it might take you quite long to get to Bergisch-Gladbach because of the rush hour and problems of the taxi to get out of the city centre.

What you could/should to is to take any S-Bahn or regional train crossing the bridge over the river to Köln-Messe-Deutz. This is one station or two minutes only and the trains are departing more or less every two or three minutes.

If you take a taxi from there it might safe you 30 minutes of getting out of town and crossing the Rhine during rush-hour. You do not need any further ticket, your long distance ticket is good for travel to Köln-Messe-Deutz too.

Enjoy your stay at Schlosshotel. You will know for sure that is has one of the best restaurants in Germany and that there is a second Schlosshotel in Bergisch-Gladbach with a comparable restaurant.