You will be hard pressed to find public phones these days, as well as computers to rent for 5 minutes for example.
You can go two ways: 1) add an international plan to your services from your current provider. Ot may be a bit more costly, bit you are sure it will work, and from day 1. 2) you can buy a Sim Card but for that your phine has to be "unlocked". In my case, unlocking cost me 30$ for my provider to punch in a code that would allow me to use a different provider when away.
If you go for the local Sim, you can purchase it is larger towns. My guess is that, assuming you are doing the Frances from SJPP, that would be Pamplona. Either is a Orange, Vodaphone or other street front store but also from their kiosks in large department stores like the Corte Ingles.
There are different packages, if you don't want to have long conversations, just to check in, a 20€ plan may work well. There are lots of companies, google a bit before leaving home. Be sure to have someone explain how to block data roaming or your plan will barely last. If your card runs out of credit, you can add to it in small grocery stores and tobaco shops.
Keep in mind that free WIFI is available in lots of bars and albergues. In bars, buy a coffee and ask for the "clave del weeefee" and you're set.
While I rarely use my phone at home I do like giving the opportunities of getting calls in case of a home emergency and vice versa, disling 112 (the European 911 equivalent), booking ahead, and I also use my photo to take pictures, its most important job.
Ok, you could also buy a phone that you will only use for the Camino, but I think that it not for environmentally friendly...
I hope this helps a bit.