This predates Nero's arrival in 2233 and does not differ from the prime timeline.
In 2263, Finnegan was in Yorktown's command center when Krall and a fleet of Swarm shipsattacked. To Finnegan's horror, Yorktown's defenses had little effect on the Swarm and Yorktown came under heavy attack.
As the attack continued, Finnegan was informed that they now had the disruption frequency and he ordered that it be broadcast by Yorktown. The combination of the signal being broadcast by both that starbase and the USS Franklin ultimately destroyed the Swarm. Finnegan was still in the command center when Scotty and Jaylah arrived to temporarily shut down the atmospheric processor. He claimed to them that what they were attempting to do was impossible. (Star Trek Beyond)
The writing of this character was heavily influenced by Simon Pegg, with fellow Star Trek Beyond co-writer Doug Jung describing the character as "a Simon thing."  According to Justin Lin, Grunberg asked to have his character be the alternate reality version of Finnegan.  Though as to why Finnegan doesn't have an Irish accent or blonde hair like his prime timeline counterpart is unknown.
Finnegan was James T. Kirk's would-be "killer" and "victim" in the titular challenge of the Star Trek: Starfleet Academy YA novelThe Assassination Game. Kirk dodged one attempt by utilizing a variation of the corbomite maneuver, claiming that he had accidentally spilled chemicals that were going to explode if Finnegan didn't leave the room immediately. In the novel, Finnegan's first name is given as "Jake".