There are two bus lines--Amtab bus, line 16, which has hourly departures, and the more frequent and more expensive Navetta, which will take you directly to the center.
But given how close the airport is to Bari, I would probably opt for a taxi rather than negotiate unfamiliar bus routes with a deadline looming. As I have never driven in Bari I can't predict how traffic conditions might affect how long it takes. If you go as soon as possible and have time to wait, the station has a couple of nice cafes, and the best train station coffee in Italy.
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