There is no standard retirement age for Shinto priests. It is also quite possible to be a priest at a jinja without actually performing many ceremonies there, so it is uncommon for priests to formally retire as priests. The comment I’ve heard a few times is that you can be an active priest as long as you can kneel formally (and get up again), but some priests remain formally in office after that, and may even lead the matsuri.
There is a title of “Emeritus Chief Priest”, which is given to priests who retire as chief priest of a jinja of some importance, and it appears that those chief priests who take that route tend to do so in their seventies or eighties.
I know this because Jinja Shinpo publishes brief obituaries of priests in every issue. (These obituaries also give the readings of their names, which is very helpful and has greatly improved my ability to read name kanji.) These obituaries give the name of jinja, the priest’s date of and age at death, and the dates of appointment to various roles at different jinja. Thus, the date of appointment as Emeritus Chief Priest is given, allowing me to calculate their age at that point.
Most issues report the death of a chief priest who had died in office, and had been chief priest of the jinja since before I was born. Last week’s issue reported the death in office of a chief priest who had held the post since before my mother was born. (He died at 98, and was appointed chief priest of his jinja in 1940, as soon as he reached the age of adulthood in Japan.)
I strongly suspect that one reason for this is the “successor problem”; a lot of jinja in Japan have no-one to take over from the chief priest when they die or retire. This means that the chief priest is unlikely to retire, because that would mean abandoning the jinja. As a result, they stay in post for over forty years.
With the chief priesthood of many jinja being hereditary, it is very common for someone to hold the post for about thirty years: one generation. If one dies young and the next lives for a long time, it is easy to get up to forty or fifty years. Seventy eight is a bit exceptional. This means that it is probably not unusual for one chief priest to perform the hatsumiyamairi (when a baby is presented at the jinja) for two generations of a family, and far from unheard of for them to do three, or maybe even four.