If you walk in the months 1 July to end of August, there is a 10am bus to Roncesvalles from Pamplona. You can take a taxi from there to St Jean for about 30€
You could also take a taxi to Roncesvalles (Pamplona - Roncesvalles € 42.00) and then hitch a ride to St Jean Pied de Port.
If not, this is what the taxi fares from Pamplona to St Jean cost.
Weekdays: Pamplona - Saint Jean de Pied de Port € 68.00
Holidays and Weekends and weekday evening Rates (22:00 PM – 7:00 AM)
Pamplona - Saint Jean de Pied de Port € 85.00
There is a cheaper - though longer - option.
You can take a train to Irun. ± 2.5 hours, costs between 15€ and 19€ and then a train to Biarritz. ± 45 mins (
http://www.renfe.es)
(There is also a bus to Irun but it leaves at 18h00 and gets there at 19h50 - costs 6,75€ http://www.laburundesa.com/ing/dinamica ... rridos.asp)
From Biarritz you take a train, bus or taxi to Bayonne and then a bus to St Jean.
At least you get to see lots of countryside!