Hi JoAnn,
I have been to Madrid many times and unfortunately it is the one city where I have been scammed 3 times by taxi drivers. There must be private car services but I have never been able to find a recommendation for one. I know of a shuttle service (aerocity.com) but for two people it would be more money that taking a cab. Having said that, I would still recommend going to Madrid and staying an extra day if you can. It's a great city. The airport is very close to the city center, aroound 20 mins in normal traffic. Regarding the cab, just make sure they start the meter. Leslie's idea of arranging a car through the hotel sounds good. Don't know why I have never thought of this. Next time I go I'll probably do that.
Same as Leslie, one of my hotel recommendations is the Villa Real. I also like the Westin Palace and the Melia ME Reina Victoria. They are all in the same area, close to Museo del Prado. That is the preferred area.
You really don't need a private guide. If you have good weather, besides the Museums, Royal Palace, Plaza Mayor, you should take a stroll on Parque del Retiro
My favorite restaurant is Posada de la Villa , near Plaza Mayor. Their specialty is roasted lamb.
Editing to add: Are you flying into Madrid the same day the cruise ends? If so, I notice it is a Sunday , so you should not have traffic issues from the airport that day.