Vanny Cache (GC1382) is the second oldest active cache in New York City, listed in July 2001. Only the infamous 4quA5 is older (by four days). They are the only two active NYC caches that were placed in 2001 or earlier.
Although it's only fair to note that "active" doesn't describe Vanny Cache very well... It's been a ghost of itself for quite some time. It hasn't been found since 2008, and by all accounts the tree that it was once located in has been more or less destroyed for much of that time. The cache owner hasn't recorded a find since 2005, and hasn't logged into the site in over a year. The listing was finally mercifully reported to New York Admin, and barring the unexpected, Vanny Cache will soon be able to happily rest in peace.
Near as I can tell, the next oldest cache is Cunningham Park #1 (GC6225) in Queens by Alan2, listed almost a year later in mid-2002. Alan is of course an active member of the community so there's probably a lot of life left in this one.