Biarritz Airport has bus service to Gare Biarritz, Pamplona, and San Sebastian. There is urban bus service from Pamplona to Roncesvalles. There is train service from Gare Biarritz to Irun/Hendaye, San Sebastian, and Pamplona.
Last bus to Roncesvalles leaves Pamplona bus station at 1800.
Last bus from San Sebastian that will get you to Pamplona on time leaves at 1600 and arrives 1700. The next one leaves at 1715 and arrives 1815. The PESA bus leaves from BIQ at 1530 and arrives in San Sebastian at 1615 ... Perhaps you can get a taxi to San Sebastian but otherwise you're not going to make it. (RometoRio site suggests 90 euro and 46 minutes) (and its still very tight for time).
Renfe train leaves Irun 1555 and arrive 1752 ... leaving 8 minutes to get to the Pamplona bus station. Train and bus station are close but not that close. And I haven't figured out how to get to Irun in time to catch the train other than taxi.
As pointed out earlier you can go the Biarritz to Bayonne to SJPdP to Roncesvalles but there is only one bus per day from SJPdP and the train from Bayonne leaves at 1810 ... and arrives SJPdP 1931 ... to late for the 1915 departure. (Check the SNCF TER site for TER#62)
So your choices as I see it are: to take a taxi from Biarritz airport to San Sebastian bus station; take taxi from SJPdP to Roncesvalles; take a taxi from Pamplona to Roncesvalles; or, take the public transit and spend a night in Pamplona before catching the bus to Roncesvalles in the morning.
Option 1.
Take route C Chronoplus bus from Airport to Gare Biarritz. Take SNCF train from Gare Biarritz to Irun/Hendaye. Or take Transdev bus from BIQ to Hendaye (hourly). Euskotren commuter rail from Irun/Hendaye to San Sebastian (every half hour).
Or take bus service from Biarritz airport to San Sebastian bus station.
There is frequent bus service provided by ALSA from San Sebastian to Pamplona. The bus station is located a couple of blocks from Amara Euskotren station. Time of bus is 1:00 - 1:30 depending on schedule.
Option 2
Take route C Chronoplus bus from Airport to Gare Biarritz. Then take SNCF train to Irun or Hendaye. Take Renfe train to Pamplona. Euskotren provides commuter service between Hendaye and Irun train station (and on to San Sebastian) if your Renfe and SNCF trains don't meet up. (3km)
Option 3
Take bus direct from Airport to Pamplona. Service twice per day.
Option 4
Urban bus #14 connects to Gare Bayonne where you can take the train to SJPdP. Take taxi to Roncesvalles. (RometoRio suggests taxi cost is 55 euro)
Pamplona to Roncesvalles:
Urban bus service to Roncesvalles is provided by Autocares Artieda and departs from Estacion de Autobuses de Pamplona. Note it does not operate on Sundays.
Taxi from Pamplona to Roncesvalles (RometoRio suggests taxi cost is 40 euro)
Biarritz Airport (BIQ) - See Services/Access and Parking/Access
http://en.biarritz.aeroport.fr/
EDSA provides bus service twice daily to Pamplona bus station.
http://www.tellevodeexcursion.com/
Conda provides bus service to San Sebastian twice per day and from San Sebastian to Pamplona almost hourly (Conda is part of ALSA). Take the bus or the train from San Sebastian. Use ALSA international tab.
http://www.conda.es/
ALSA
http://www.alsa.es/en/
PESA provides bus service to San Sebastian twice daily. Take the bus or train from San Sebastian.
http://www.pesa.net/pesa/public_home/ctrl_home.php
Transdev ATCRB
http://www.transports-atcrb.com/
SNCF
http://www.sncf.com/en/passengers#
SNCF TER Acquitane - train Bayonne to SJPdP is TER 62 - train Biarritz to Hendaye is TER 61
http://www.ter-sncf.com/Regions/Aquitaine/fr/Default.aspx
Renfe
http://www.renfe.com/index.html
Euskotren
http://www.euskotren.es/
Estacion de Autobuses de Pamplona
http://estaciondeautobusesdepamplona.com/Inicio.aspx
Autocares Artieda
http://www.autocaresartieda.com/?scc=svpropios
Rome to Rio
http://www.rome2rio.com/