From Zariquiegui you could walk an extra 2 km cross country to Astrain, on what looks to be a flat paved trail. At Astrain you can catch a bus to Puente la Reina. The La Estellesa bus services that route.
https://taquilla.laestellesa.com/billete/listado-billetes#
If you are wanting to avoid Alto del Perdon for physical or time constraints, another option to get to Puente la Reina, and still walk some of the way, would be to get the La Estellesa bus from Pamplona to Legarda. The bus doesn't actually go into Legarda, but lets you off on the highway. From there you can walk into Legarda and then cut across country (about 1.5 km on farmland trails) to Uterga and proceed on to Puente la Reina. I did this in 2022 and also took the detour to the lovely Church of Saint Mary of Eunate.