Bilbao to SSJP?

PESA bus from Bilbao Loiu Airport to Donostia/San Sebastian.

If you arrive late take local bus from airport to Bilbao Termibus

Then take ALSA bus from Bilbao to San Sebastian.

Depending on your timings:

From San Sebastian outdoor bus station located at Plaza Pio XII

PESA bus from Donostia/San Sebastian to Place de Basques Bayonne; or,

CONDA (ALSA) bus from San Sebastian to either SJPdP via Pamplona or TGV Bayonne (which I believe but have not confirmed is ALSA description for Gare de Bayonne) (note last bus to SJPdP leaves Pamplona at 1730); or,

Metro Donostialdea from San Sebastian Amara to Hendaia. Amara is located about 700 meters from Plaza Pio XII. Hendaia is colocated with Gare d'Hendaye.

From Hendaye

SNCF train from Hendaye to Bayonne; or,

Transports64 bus 816 'Euskadi Express' from Hendaye to Place de Basques.

From Bayonne

Place de Basques is located about 1 km from Gare de Bayonne. Walk or take local Chronoplus bus.

Take SNCF TER 62 train from Bayonne to SJPdP (note last train usually leaves 1806 (or 2110 on Sunday)).

And if none of this works out try Private Hire Taxi: Express Bourricot who will pick up by prearrangement (if available) from Hendaye or Bayonne.