I think that if Robert had no money he'd still be arrogant and self-centered. He'd just find something else to be arrogant about.

He strikes me as a person with an over-developed sense of self, and a snobbery that will carry through regardless. When Elinor first meets him he has no settled income - that only comes when Edward's engagement is open. Yet Robert is snobbish and self-centered even so, with all his stories about holding balls in cottages and the like.