I’m not close to my reference sources (at home) so I can’t answer your Q outright, but I can put a little context to your question.
One thing you must keep in mind that NYC has not always existed in the state that we know it today. The city of Brooklyn and the various towns and villages in Queens, Westchester, and Richmond (S.I.) Counties that got absorbed into NYC (in 1874, 1895 and 1898) had primary public schools too, and some of them may be older than the ones that currently exist in Manhattan. (The whole system got unified in a citywide BOE in 1898.) Brooklyn, in particular, had a very strong education system; Erasmus Hall (of Woody Allen and Barbra Streisand fame), for example, long predates any public HS in Manhattan.
Also, residential lower Manhattan has undergone many “pave-overs” since the mid nineteenth century. There are not too many kids living on Wall Street! In fact, the financial district was, until recently, not zoned for residential living at all.
Oh, and I’m pretty sure (but not absolutely certain) that the school numbers do not relate to founding dates.