How would he know that since he didn't have you there to tell him?
He rose to prominence in New York Politics through his work on two African American Civil-rights cases in the 1850’s. The 14th Amendment was also passed just a few years before Arthur would get the Nomination. He was a lawyer, and was well versed in issues of citizenship.
Now consider that the day his father got Naturalized as an American, given the importance of citizenship, and of attaining it, it was then and still is today a BIG deal, at 14, Chet was sure to have known about it. Chester knew he wasn’t a Natural Born Citizen, but because his family was grounded in the states for a long time, no one would have though to look there, and Chester himself could bent the truth, something he was willing to do, and state truthfully that his dad was a Naturalized citizen, he just left out WHEN he became Naturalized. A half truth, and people bought it. It was believable. That being said, Chester was still of British and American Citizenship, and not a Natural Born citizen and he knew it.