The aura from the rats is an interrupt attack at the beginning of the pc's turn. I would rule the ongoing damage damages the pc that very turn.
Except the attack isn't an immediate interrupt, it's a free action. The only "step" it makes sense in is "other effects," in which case whether you believe the list on p. 268 is the specific order of operations or that the player can choose the order is irrelevant, because "other effects" are lower on the list than "ongoing damage" and thus you've already taken your ongoing damage for the turn by the time the rat swarm bites you, or the player chooses the order and chooses to resolve other effects after ongoing damage.