The closest airport to SJPdP is Biarritz/Bayonne
http://en.biarritz.aeroport.fr/
The BIQ website does not indicate flights from Budapest ... so you would need to travel via an airport that connects to BIQ.
There is Chronoplus bus from Airport to Gare Bayonne
There are SNCF trains to SJPdP from Bayonne, usually 4 x per day, all year.
Travel via Biarritz is probably the most convenient.
There is CONDA (ALSA) bus service from Pamplona to SJPdP, currently once per day (starting 1 March 2016) with more buses added as peak season draws closer.
There is frequent bus (Vibasa direct from SANTS and ALSA from Barcelona Nord via Zaragosa) and Renfe train service from Barcelona to Pamplona.
SNCF Train from Bordeaux or Toulouse to Bayonne and onward to SJPdP would likely be faster than travel via Barcelona and Pamplona ... however, your schedule might or might not work out that way. You'll have to do the time appreciation yourself. Cost is an important factor as is determining how you will get home.
Many of us here are traveling from points across the globe. Most of us don't have access to flights that will go to Bordeaux or Toulouse without first arriving at a more major center such as Paris, London, or Madrid at which point travel times and costs are competitive using rail and bus versus flights.