The Predator in Prey doesn’t survive to become one of the aliens we saw in that film; Naru kills it in Prey’s final scenes.
The traders slaughter a pack of buffalo and later capture Naru to use her as bait against the Predator.
The object in question is an old flintlock pistol, which becomes an important weapon in Prey’s battle between a new Predator and a young Comanche warrior named Naru, who is the film’s hero.
It remains unclear how the pistol went from Naru back to the Predators so that they could then give it to Harrigan centuries later.