Google Gray Line Tours and book the hop on-hop off double decker shuttle. Don't get off and do the South Tour, you'll see Times Square, Empire State, Chinatown, SoHo, City Hall, World Trade Centre Site, Wall Street. Get off at Battery Park and take the Staten Island Ferry(Free!) and you'll go past Statue of Liberty and Ellis Island, Richard is right to do this on its own would take up too much of your time. On arrival at Staten Island get back on board for great photos of South Manhatten. Pick up bus again and you'll see South Street Seaport, UN building, 5th Avenue, Rockefeller Centre, Broadway and back to Times Square. If you have time go up the Rockefeller Centre tower rather that the Empire State, much better views and hardly any queues. Have a great time!
Yes, taxis will be available at the Brooklyn Terminal, however, there is likely to be a bit of a scramble, particularly if it is wet. A taxi to Manhattan will take 20-40 minutes and cost approx $25.Once through US Immigration, if you find queues for the taxis, head off following the exit road for approx 200 yards. Before the road bears left there is a small pedestrian gate in the fence, exit through and head for the main road where you can get a bus to the nearest Brooklyn Subway station. I'm pasting the following for you which I copied this week from another contributor, looks sound but I have not tried myself : "use the very safe NY subway. It costs $7.50 for a day's unlimited travel, then catch the free water taxi from pier 12 at Manhattan to Ikea at Brooklyn. The cruise terminal is only 15 minutes walk away)". Hope this helps. You will not achieve all the major sights in such a short timeframe but many will be . I do not think you will have time to visit the Statue of Liberty or Ellis Island-the queues are very long and start too early. If all else fails Cunard usually organise a full day tour with the odd hour free for shopping and you have the comfort of knowing you will get back in good time. Have a lovely time, sounds fantastic. Richard.