So the "oldest building" part is a tad bit misleading. It should really be the oldest building of non-indigenous origin, ie. European. But I digress. The oldest building in the western hemisphere is actually an ancient Spanish monastery that was shipped to the United States in pieces and then rebuilt eventually in North Miami Beach.
The monastery feels rather odd set against the modern hotels and skyscrapers of South Florida, but it is a pretty cool building and a nice way to see some Spanish history without buying a plane ticket.