I'm not finding much. I have found a bus from Bilbao to Pamplona (2 hours, about 13 euro one way, one bus leaves at 6, the closest to your arrival time), but nothing from Pamplona to Estella online. I *know* there are buses to & from Estella because I've seen the bus station & gone through there on my way to San Sebastian from Logrono.
My advice would be to overnight it in either Bilbao or Pamplona. You may be able to find an early morning bus from Pamplona to Estella, so you can still start your walk the next day.
Kelly