Archer's duties including teaching at Starfleet Academy, where he taught the Advanced Relativistic Mechanics course. During his time at the Academy Montgomery Scott served as the aide to the Admiral on this course. (TOS website: Dossiers)
Scott once used Archer's "prize beagle" (unaware it was the Admiral's) in a transporter experiment to try and prove his theory on transwarp beaming. Unfortunately the beagle did not rematerialise. Scott subsequently blamed this incident for his being transferred to the Delta Vega outpost, which he considered a punishment - as he recounted to James T. Kirk (who was also familiar with Archer and the beagle) and Spock in 2258.
The Admiral's name, and fondness for beagles, are a clear reference to Jonathan Archer from Star Trek: Enterprise. Indeed one of Star Trek's writers, Roberto Orci, has commented that the reference is intended to be that Archer. However this information is not given in the film, and the Archer could just as reasonably be a descendant, or another Archer (with a coincidental fondness for beagles) altogether. Furthermore Jonathan Archer is established to have already risen to Commander in Chief of Starfleet by 2168, when he retired to follow a political career, eventually becoming Federation President (ENT episode: "In a Mirror, Darkly"; ST novel: Mere Mortals), which would seem to stretch the plausibility of him returning to Starfleet in a minor role much later in his life (age 146 in 2258).