I'm still trying to get over the fact that my honorary grandfather is
Is there any explanation of what happened to his character in Star Trek: Beyond?
As you note, early in Star Trek Beyond, the Kelvin-timeline Spock is made aware that Prime Spock, who we last saw in Into Darkness on New Vulcan, has passed away.
The cause of death was not made explicit in the film. The script notes do not shed any further light on this. Also, note that an official novelization of Star Trek Beyond was never released.
Presumably, he died of old age, given that he would have been 162 at the time (based on his official database entry at StarTrek.com). This is not exactly of "advanced age" for a Vulcan (whose lifespans can meet or exceed 200 years, as evidenced by Sarek), but is certainly advanced for a human. As Omegacron notes, given that Spock is half Vulcan, half human, one might surmise that 162 is an advanced ge.
Finally, we should note that Spock's passing was a later addition to the script, and that it was a direct response to Leonard Nimoy's passing in early 2015. According to Beyond director Justin Lin:
LIN: It's something you'll see in the film [a reference to Nimoy's passing]. It obviously affected everybody because he's been a big part of our lives. There's an attempt to acknowledge that in some way.
As Spock's death was meant to parallel Nimoy's passing, one might surmise that Spock died of natural causes, like Nimoy himself.
Spock Prime died, presumably, of old age. When Spock Prime died on New Vulcan, he would have been approximately 162 years old.
Sarek, Spock's father, was considered to be middle-aged at 102 (TOS) and very old at 203 (TNG). Using this as a scale, we can assume that 162 for a Vulcan would be roughly 80-ish for a human. And like humans, not all Vulcans could expect the same lifespan - all manner of criteria, including genetics, would factor into determining their natural lifespan.
Couple this with Spock's half-human biology and we can safely assume that 162 was a pretty good lifespan for him - just a couple of decades shy of the Vulcan norm, in fact.
Spock Prime dies at the start of or just before the events of Star Trek: Beyond.
When Enterprise arrives at the Yorktown space station, Spock is briefed by Vulcan emissaries about the ambassador's death, news that influences Spock's actions throughout the rest of the movie and his decision whether to remain in Starfleet.