In question 122 of Mugimeshi-san's question corner, somebody asked whether the fairy king's lifespan is longer than the rest of the fairies. Which gets answered with "It might be the longest out of all the different species."
But how old can these fairies become?
As of now, there's no official answer, but at least older than 4200 years.
Summarizing from the Wikia about Fairy Clan:
Longevity: The average lifespan of a Fairy is between 1000 to 1500 years. They do not appear to physically age and they never suffer from illness.
Now, comparing the known fairies' age:
Although most of them are mentioned in between 1000-1500 range, Gerheade is an exception, and she's not even a king.
Thus, assuming there's nothing that prevents their natural aging (e.g. Elaine, Helbram), and since they don't suffer any illness, fairy king should even be possible to live more than 5000 years.