Hi beacher. You could try by bus. In the same webpage www.alsa.es you can buy tickets from Madrid to Pamplona and from Pamplona to SJPP. Quite expensive this last trip (around 22 eur) but more suitable.
There are some services to Pamplona from Madrid airport.
Hope you a nice Camino.
I will also start from SJPP on 6th of August. Since I will arrive very late I suppose I will spend the night resting in Valcarlos.
As the other guys told, you will find very nice people on the way.
Buen Camino!
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