Yes. There are buses from the airport to the Gare SNCF Bayonne throughout the day. Journey time is less than 1 hour and you can buy the ticket at the bus.
From Bayonne to SJPdP there is presently a local train/bus combo. For this trip you purchase a ticket (roughly 10 euros) at the Bayonne railroad station (Gare SNCF Bayonne). The same ticket will cover both travel on the TER train and travel on the TER supplied bus. Total travel time on this train/bus combo is roughly 1 and 1/2 hours.
This current
2015 TER Aquitaine line 62 schedule shows the combination of train from Bayonne to Cambo les Bains and TER supplied bus from Cambo les Bains to SJPdP as being the travel combo until
July 4, 2015. The schedule for later travel in 2015 has not yet been issued.
Your date June 17 is a weekday with a train/bus combo scheduled to leave Bayonne at 18:06 arriving SJPdP at 19:24.
Bon voyage and Buen camino,
MM