No. "Priesthood" is a noun meaning the collected ordained priests of a particular religion. "Priestliness" is a back-formed noun derived from the adjective "priestly" which means "like a priest." A person who is non-religious may behave like a priest and therefore have priestliness but not be a member of the priesthood.
Priestliness is an awkward, cobbled-together word that I would never use. If I wanted to describe priestliness in a negative way I would use sanctimony, if I wanted to describe it in a positive way, I would use piety. Neither of those words mean "like a priest," but then not all priests are alike. If I wanted to say that someone behaved like a priest and that behavior was annoying, I would say that I'm annoyed by his priest-like behavior.