Last September after crossing the pass, we were in Jaca, a magnificent town. Feeling a bit sick, we decided to take the bus to Santa Cruz de la Seros. This is the bus that goes once each morning all the way to Pamplona, stopping at numerous towns along the way. I can't remember the name of the bus company, but the bus departs every morning. While we were waiting for our bus, we noticed a van (shuttle bus) loading up a passengers headed for the monasteries of San Juan de la Pena. It left within 15 min or so of our bus and went directly to the monasteries. We spent the night in Santa Cruz and hiked up to the monasteries the next morning. We were passed by the same shuttle bus as we climbed. Since it ran two consecutive days, I assume it runs every day in the summer. Both the main bus and the shuttle depart from the main bus station. I am sure the tourist information office in Jaca could give you exact information for both bus options.
The main bus lets you off for San Juan on the main highway. It is still about an hour's walk to San Juan.