There was no reason to make Spock a full commander at the age he was in TOS when he was a lieutenant either.
Actually Spock was a lieutenant in "The Cage" in 2254, whereas this movie is set mainly in 2258. There's certainly precedent for going from lieutenant to full commander within four years. Will Riker was a lieutenant in 2361 (according to "Peak Performance") but a full commander by the time he joined the Enterprise crew only three years later. And Geordi LaForge went from lieutenant J.G. to lieutenant commander -- an equivalent two steps in rank -- in just over one year, from the end of 2364 to the start of 2366.