There is a direct bus from the bus station in Bilbao to the bus station in Pamplona. La Unión/Burundesa. (And there is a frequent bus from airport to bus station). When I travelled over in June from Dublin, my plane was too late for me to get my bus at 6pm so I was very lucky to find a car using blablacar. It was a great trip, and I was able to walk from where the driver left me to the house I was going to. I just fished around and joined blablacar the day before travelling. It seems secure, with transparent comments on drivers and cars.. and it is slightly cheaper than the bus, although that is not really the point. Wishing you all the best, and another site for timetables is rome2rio, or something like that. Buen Camino!
We've just done this. There's local buses from the airport to the city bus station in Bilbao every 15 mins and the journey takes about 20 minutes costs about €2. Then take La Union bus to Pamplona direct, costs €15 takes 2 hours.