Apparently he went back to Qo'nos at 15 do do his Klingon adulthood rites
Probably more like 8 or 10, about the same age Alexander was when he joined the Rotarran. Old enough to make a choice, but young enough that he still went back to Galt when it was over.
In "The Icarus Factor," Worf celebrates the tenth anniversary of his Rite of Ascension. Since it's the second season, that makes him about 25, so his actual Rite of Ascension would have taken place at about 15.
I don't think Alexander would have gone through that before joining the Rotarran. Perhaps some sort of preparation is necessary before doing it - like service on a ship, or going on a hunt, or whatever (Worf killed a kid playing soccer; maybe that counts
). Plus, looking at the rest of the new recurits that episode, the Klingons were hurting for people, and would take just about anyone willing, adult or no, just as long as they were halfway competent (and only halfway, as Alexander's performance showed