That would depend on the CIWS. Most modern anti-ship missiles are armoured to a point where a gun such as the Vulcan used in the Phalanx system has more than a little trouble denting them, but the more recent missile based systems (like the ones based on the RAM Mars mentioned) would fare better. They're also often programmed to fly erratically when getting close to the target, letting them avoid much defensive gunfire - Again, though, that's less of a problem for a missile based system, since they can track it in flight.
Still, even the best CIWS in the world will not have a perfect record against something like the Sunburn. It's too fast, too armoured, and, if other Russian missiles are any indication, probably too smart for certainty in interception.