I would not get into any NY taxi and expect a tour off them. Most of the ones I have encountered have only a rudimentary grasp of English.....I would suggest mykindoftownny.com as a good staring place.
I would agree, it would cost you an arm and a leg anyway. Manhattan is easily reached from QM2 docks and with a little bit of forward planning work out what you would most like to see and plan a route….Nei.
An alternative might be to pre-book a limousine ride. I would not suggest hiring one for 4 hours though as this would work out really expensive. We hired one to give us a drive around the main sites for about 90 minutes and it was very good. It dropped us off at a restaurant in greenwich Village and we just grabbed a yellow cab back for the return journey. There are quite a lot of things to do and see that are walking distance form 42nd St/Broadway, so laternatively just get a cab from the airport to there and have a wander about before getting a cab back. There are several hop on hop off bus routes in NY and whilst you would not have time to get on and off, you could have time to do one loop just to give you an impression of the cit.
Regretfully I must agree with Aldrich. New York taxis bear no resemblance to London ones. Usually the drivers have no knowledge of the city. Rarely, when they can speak English, they push rudeness to the extreme. Even a chauffeur car booked for me by BA Concorde had a driver who refused to communicate. The best of luck.
Have only just seen your question. Note that you are in Bebington - presume this is Wirral, u.k. We live in Great Sutton and are on the same cruise as you. We are about to book a limousine for the journey from ship to airport. Our flight isn't until 6pm so we are going to have rather a long wait at the airport. Wonder if you have any ideas about sharing etc. If you would like to contact us direct our email address is email@example.com and we are leaving here early Thursday morning. We are in Cabin 6266 on board if you would like to contact us there.