The nearest airport is Bari-Palese, see the Bari Airport web site. A shuttle service will be organized from the Bari airport to Matera on June 24th, and from Matera to the Bari airport on June 28th.
The Bari-Palese airport is connected every day to Matera by buses of Puglia-airbus. The timetable can be found here
Matera is connected to Bari by trains and buses of Ferrovie Appulo-Lucane.
On Sunday, buses leave from via Capruzzi 236 - Bari (100 m from the Bari Centrale railways station) with the time schedule:
8:10 am - 10:05 am - 1:00 pm - 4:40 pm - 6:30 pm - 7:15 pm - 8:40 pm.
Buses are of blue or blue-green color, with the inscription "Ferrovie Appulo Lucane" or "FAL". The destination "Matera" is indicated (if you are not sure, ask the driver). Bus tickets can be purchased online at this wabe page or at the travel agency "ATS viaggi" at via Capruzzi 224/C-226. The bus stop is in correspondence of the agency on the opposite side of via Capruzzi.
Do not forget to notify us your expected arrival time, so that the we can arrange the shuttle service on June 24th. Private taxi service would be expensive, since the Bari airport is about 60 Km far from Matera.