Just take the train from the airport to Victoria Station. There is a train from Victoria Station to Dover. I remember it taking about 90 minutes or so. I think it was about 80 quid (pounds) (roughly $120) round trip for two, as I remember. You can use your visa or mastercard to purchase your train ticket and get the very best exchange rate. That was in 2000, so prices may have changed somewhat. I doubt they have changed a great deal, however.
There are elevators in Victoria Station. Don't be stupid like us, and carry your bags up the many flights of stairs from level to level.
In Dover, be sure to take the double decker bus tour around town and do spend some time at the castle. It is beautiful and most interesting. There are a lot of B&Bs very close to the train station in Dover. Often the proprietors of the B&Bs will take you to the cruise terminal in their personal cars. Otherwise cabs are available.
There are some wonderful pubs in Dover. Please have a pint for me. Cheers!
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