I would estimate you get as far as Burgos.
The closest airport to SJPdP is Biarritz
Biarritz Airport (BIQ)
http://en.biarritz.aeroport.fr/
AENA manages airports in Spain. Madrid and Bilbao are airports with flights to Dublin.
http://www.aena-aeropuertos.es/csee/Satellite/HomeAenaAeropuertos/en/Home.html
ALSA provides bus service from Burgos to Madrid and Burgos to Bilbao.
http://www.alsa.es/en
Renfe provides train service from Burgos to Madrid and Burgos to Bilbao.
http://www.renfe.com/
Autobuses Jiminez provides bus service between Logrono and Burgos.
http://www.autobusesjimenez.com/
Autobuses Amaya provides bus service from Fromista and Castrojerez to Burgos.
http://autobusesamaya.com/
1000 Saint Jean Pied de Port
1001 This section discusses transportation services in Saint Jean Pied de Port (SJPdP)
Index 1000
1002 Saint Jean Pied de Port
1003 Saint Jean Pied de Port to Bayonne
1004 Saint Jean Pied de Port to Pamplona
1005 Saint Jean Pied de Port to Pau
1006 Private Hire Taxi and Luggage Transport
1007 Pilgrim's Office
1002
Saint Jean Pied de Port
Saint Jean Pied de Port website
http://www.st-jean-pied-de-port.fr/default.aspx
1003
Saint Jean Pied de Port to Bayonne
SNCF TER Aquitaine provides train service from SJPdP to Bayonne, all year. The train is designated TER 62. At times a bus replaces the train and runs on the same schedule with the same stops.
At time of writing tracks are washed out so the train only goes from Bayonne to Cambo les Bains. An SNCF operated bus continues from Cambo les Bains to SJPdP. The schedule is changing often.
SNCF TER
http://www.ter-sncf.com/
Transports64 provides bus service to Pyrenees-Atlantiques Department including Bayonne to SJPdP via St Palais. See schedules for #811 and #810.
Transports64
http://www.transports64.fr/transports-64?lang=fr
1004
Saint Jean Pied de Port to Pamplona
CONDA (a subsidiary of ALSA) provides ~seasonal~ bus service from Pamplona to Saint Jean Pied de Port. The bus makes a stop in Roncesvalles and Valcarlos. The SJPdP bus stop used by CONDA is located near the junction of Route d'Uhart and Route d'Arneguy. Trip time is one hour 45 minutes. The bus then returns from SJPdP to Pamplona. The CONDA site advises that you purchase the ticket from Pamplona to SJPdP separately from any other bus ticket.
Use the ALSA schedule.
The bus service usually starts just prior to Easter and terminates at the end of autumn. Schedules are not usually updated more than one month in advance.
The bus from Pamplona to SJPdP ran daily 1000 13 June to 31 August 2014, 1430 1 June to 30 Sept 2014, and 1730 17 March to 30 September 2014. The schedule for the 1430 bus was extended first to 3 Nov 2014 and again to 7 Dec 2014.
ALSA
https://www.alsa.es/
Conda
http://www.conda.es/
1005
Saint Jean Pied de Port to Pau
Transports64 provides bus service from Pau to Saint Jean Pied de Port. Several transfers are required. One option is 801 Pau to Orthez, 809 to Saint Palais, and 810 to SJPdP.
Transports64
http://www.transports64.fr/transports-64?lang=fr
1006
Private Hire Taxi and Luggage Transport
Express Bourricot, based in SJPdP, offers 'person transportation' service to and from any location in the region including Biarritz Airport. This service must be arranged in advance. The cost goes down as the number of passengers goes up.
Express Bourricot operates a 'luggage transportation' service to Roncesvalles and Santiago.
Express Bourricot
http://www.expressbourricot.com/
1007
Pilgrim's Office
The address of the Pilgrims' office in SJPdP: Amies du Chemin de Saint Jacques at 39 rue de la Citadelle. Look it up on google earth and make a copy of the map. SJPdP is a small town but the streets are confusing, particularly if you arrive after dark.