No. Arrow's theorem does not depend on "the inputs" being strict rankings (orderings), as opposed to partial rankings. It is common for proofs of Arrow's theorem to only take into account strict orderings, but that's just a simplification and is done without loss of generality. So yes, Arrow's impossibility result still holds if we allow partial and incomplete orderings, which is the point against Mr. Poundstone's claim that "scored" methods bypass Arrow's theorem.
If you still don't get it, try thinking about it this way. If you can prove that white cats are not immortal, you've automatically proven that cats (generally) are not immortal. Similar reasoning is used in Arrow's theorem. Arrow showed that there is no social welfare function that satisfies a set of desirable conditions; he did this by showing that there are no social welfare functions that, when "fed" strict orderings, satisfy those conditions; since there are no social welfare functions that satisfy these conditions, at least when fed strict orderings, there are no social welfare functions that satisfy these conditions regardless of what they are fed.