Is the dog Porthos?
I include this section because I didn't initially believe it could be the same dog, but research proved me wrong.
Robert Orci, one of the writers of Star Trek (2009), answered this question in a 2009 Q&A on TrekMovie.com1:
QUESTION: Forget with it was Admiral Jon Archer (as it obviously was!): is the beagle mentioned Porthos? Has veterinary science advanced so much? (Please say yes!)
Orci: YES!
So shut me up.
How old would Porthos be?
We don't know exactly when Archer first got Porthos, although we know he's had him from a puppy. "Broken Bow", the first episode of Enterprise, is established as taking place in 2151:
Archer: Enterprise starlog, Captain Jonathan Archer. April 16, 2151.
Star Trek Enterprise Season 1 Episode 1: "Broken Bow"
So Porthos has been around at least since April 2151.
Meanwhile, in Star Trek (2009), Spock gives the stardate after taking command of the Enterprise:
[T]he Enterprise BLASTS AWAY -- in the distance, THE ENTIRE PLANET OF VULCAN SWALLOWS INTO ITSELF -- A TERRIFYING, UNIMAGINABLE IMPLOSION -- UNTIL NOTHING IS THERE BUT DARKNESS. Over this, we HEAR:
SPOCK (V.O.): Acting Captain's log. Stardate 2256.42: I have assumed command of the Enterprise
Star Trek (2009) - 2007 draft
According to Izkata's answer to this question, that stardate corresponds to February 11, 2256.
Later dialogue from Scotty indicated that he made Porthos disappear six months prior:
Starfleet Officer (cont'd): Six months I've been living on nothing but Starfleet Protein Nibs and the promise of a real food delivery!
[...]
Scotty: I told my instructor I could not only beam a bird from one planet to an adjacent planet in the same system -- which is no big deal anyway -- but if I were so inclined I could actually do it with a lifeform! So I tested it on Admiral Archer's prize beagle. Which... was a mistake
Star Trek (2009) - 2007 draft
Six months before the above date puts us around August 2255.
So Porthos was at least 104 years old.
1 The questions are asked and answered in the comments under the blog post. I've tried to permalink to Orci's reply, but may not have succeeded