Most hotels offer some kind of Internet service; if yours doesn't, the front desk should be able to direct you to one of the small Internet cafés In St. John's, Jolly Harbour, or English Harbour (the operations move around quite a bit).
Airmail letters to North America cost EC$1.50; postcards, EC75¢. Letters to the United Kingdom cost EC$1.50; postcards EC75¢. Letters to Australia and New Zealand are EC$1.80, postcards EC90¢. The main post office is at the foot of High Street in St. John's. Note that there are no postal codes; when addressing letters to the island, you need only indicate the address and "Antigua, West Indies."