I have also looked at the website Laura mentions. A lot of those services are historic details of last summer's routes. However, I did find some that worked at the moment. But to repeat laura, please don't rely on me, either!
There is a bus M-F at 1300 to Larasoana, (the Euqui service)
The 1600 Saturday bus and 1800 Weekday bus (Jaurriete) also stops at Larasoana, Zubiri, Linzoan, Viscarret, Espinal, Burguete as well as Roncesvalles. So you could decide to go as far up the road as looked walkable.
If you get to Roncesvalles and think better of it, there are buses reversing the route of the 1600/1800 service towards Pamplona at 09.20, but only Mon - Fri. Looks like you will be there over a weekend, like me. So we don't have this "cheat" option.
Rather than having to make a fast decision after your flight on Friday, you could stay overnight in Pamplona, use the morning to sightsee, then take a view at lunchtime about either setting off to walk for a short first day, or getting the 1600 bus either to Roncesvalles or Larrasoana. The ride would be in daylight so you could see what you were getting into and being two hours earlier you might have a better chance of getting a bed than if you were catching the Friday bus.
Sorry - some unsolicited advice - mostly musing about what I shall do in a week's time.