No, the spacecraft just has a bunch of extra fuel and is going to use slingshot trajectories to catch up to Apophis after the asteroid's Earth encounter. Though it will be close enough at that point to accurately measure the asteroid's position to find how it will be perturbed by the gravitational encounter with Earth. When Osiris-REX reaches the asteroid it will use its thrusters to ablatively remove the top layer of the surface to get a rough idea of its internals.
Galaxy brain time, Osiris being the Egyptian god of fertility, agriculture, the afterlife, the dead, resurrection, life, and vegetation chasing down Apophis, the Egyptian god of chaos is pretty neat.
