John,
From the looks of it, it seems probable the early members laid out the first holes over the winter of 1897/98, of which there were nine in play by 1901. The fact that the club was organized in October 1897 and by November they were trying to prepare a course (which they had in play by March 1898) tells me that it was a crude one put together by the golf members of the club most likely, probably Van Syckel, Roebling, and Janeway. A May 1898 article I found indicated that golf was becoming more popular than tennis.
Jimmy Norton's 18 hole course in 1913 was termed a "new" course, but it's likely that he used at least some of the original holes.