I've never flow into the Pamplona airport, but I've researched the issue, and I don't think that's accurate. According to the airport's website, there is only taxi service directly to the airport.
https://www.aena.es/en/pamplona.html. (The City of Pamplona website indicates direct bus service was suspended in 2019).
However, if you don't want to take a taxi, you can walk approximately one kilometer from the airport and pick up local bus L16, which will take you directly to the Pamplona Bus Station. Buses run approximately every 15 minutes, and the ride is another 15 minutes. Here's the schedule from the web site for the bus company:
https://infotuc.es/index.php/es/horarios-y-recorridos/linea/16.
Google Maps will show you the way to the bus stop. Input Palermo Airport as the starting point and Palermo Bus Station as the destination, then choose the Public Transit icon (it's a train symbol) and it will show you the walking directions, plus the bus's route into Palermo. (Rome2Rio will show you the same thing, if you drill down to the detail on its map).