There isn't a causal mechanistic link between shingles and Alzheimers. We don't know why the Shingrix vaccine reduces incidence of Alzheimers.

(You should very much want to avoid any chance of getting shingles, though; it's a nightmare.)

The shingles virus hides in the nervous system and can come back out when immune control weakens. It causes inflammation and nerve injury, sometimes severe enough to leave people with chronic pain. From there, it’s reasonable to theorize that it could also affect the brain and even its glymphatic system over time.