Option 1
Conda bus from Biarritz BIQ airport to Pamplona bus station. La Estellesa bus from Pamplona bus station to Estella.
Time appreciation:
Arrive BIQ airport 1255
Conda operates buses from Biarritz Airport leave 1410 arrive Pamplona 1615 (next bus leaves 2030 and requires stay over at Pamplona)
La Estellesa bus leave Pamplona 1630 arrives Estella 1720 (next bus leaves 1800)
Biarritz Airport
http://en.biarritz.aeroport.fr/
ALSA owns Conda ... use the international tab
http://www.alsa.es/en/
La Estellesa
http://www.laestellesa.com/
Option 2
ATCRB buses go from the airport BIQ to Gare d'Hendaye almost hourly. Euskotren operates a commuter train from Hendaia to Colon station in Irun. Cross the bridge over the tracks and turn left to find the Irun Renfe Station. Estellesa buses run two times a day from the Irun Renfe Station to Estella via San Sebastian. The last one leaves Irun at 1545.
Time appreciation:
Arrive BIQ 1255
ATCRB bus leaves airport 1346. The next one leaves at 1501 and arrives too late for onward travel.
Arrive Hendaye 1457
Euskotren Hendaia 1503 or 1533 (the 1533 is probably too tight ... walk or take a taxi to Irun Renfe station if you miss the 1503)
Arrive Irun (Colon) 1507 or 1537
Estellesa Bus 1545 (last bus)
Arrive Estella 1800
This time appreciation depends on flights arriving on time ... It will work but there is no margin. If you go this route you might want to plan for a night in Irun or San Sebastian.
BIQ airport
http://en.biarritz.aeroport.fr/
ATCRB bus
http://www.transports-atcrb.com/
Euskotren
http://www.euskotren.es/
La Estellesa Bus
http://www.laestellesa.com/
Option 3
Bilbao airport to Bilbao bus station. Autobuses La Union to Logrono and La Estellesa bus to Estella.
Bilbao Termibus
http://www.termibus.es/index.php?idi=es
Estellesa operates buses every 2 or 3 hours between Logrono and Pamplona with Estella being the midway point.
http://www.laestellesa.com/es/horarios-compra
Time appreciation
Arrive Bilbao Airport BIO 1535
Local Bizkaia Bus leaves airport 1610 arrive Termibus 1640.
Note the timing is 35 minutes to deplane and catch the bus. The next bus arrives at 1710 ... which is too late. If you miss the 1610 bus you ~might~ make it with a taxi.
Autobuses La Union leaves Bilbao 1700 arrive Logrono 1845
La Estellesa bus leaves Logrono 1900 arrive Estella 2000 (last bus)
This time appreciation is very tight. I'm not sure you can get from the plane to the bus station in time. If you go this route you might want to plan for a night in Logrono.
BIO airport
http://www.aena-aeropuertos.es/csee/Satellite/Aeropuerto-Bilbao/en/
Bilbao Termibus (for airport bus and LaUnion bus to Logrono)
http://www.termibus.es/index.php?idi=es
La Estellesa bus
http://www.laestellesa.com/
There are other options including travel from Biarritz or Bilbao to San Sebastian and onward to Pamplona.
-PESA bus from Bilbao (Loiu) Airport to Donostia/San Sebastian. Last bus leaves airport 1645.
or
-ATCRB bus from Biarritz airport to Hendaye (hourly basis)
-Euskotren from Hendaia to San Sebastian Amara (every half hour)
-Amara to San Sebastian bus station at Plaza Pio XII 700 meters ... walk or taxi
Then
-Conda (ALSA) bus from San Sebastian to Pamplona (hourly basis)
-La Estellesa bus from Pamplona to Estella ... (every 2 or 3 hour - last bus leaves 2030)