There are several ways to get to Bari. The Bari International Airport-Karol Wojtyla is where you want to fly into. Then its an hour and 15 mins to Ostuni. https://www.rome2rio.com/map/Bari-Airport-BRI/Ostuni
Italy uses the Euro. Best to exchange US Dollars for Euros prior to getting to Italy.
Make sure your passport is up-to-date and allow time to renew. It usually takes 3-9 weeks depending on how you process it.