Specifically, during British rule before the American Revolution. Were houses numbered or just referred to by the surname of the resident? And was this true for all houses, even the houses of the poorer inhabitants? How was this determined?
This was an interesting question to me that I unfortunately could not find a definitive answer to, but I did come close. Here's what I was able to turn up:
A collection of the New-York Gazette, the city's second newspaper, published between 1725 and 1744 turned up no mentions of house numbers. When people's houses were mentioned, it would say things like, "Sold by Benjamin D'harriette at his house in Wall Street" (1). Or "John Wallace, who lately lived by the Old Slip Market, is Removed to the Sign of the Cross-Roads, next Door to Mrs. Byfield..." (2).
A collection of Rivington's New York Gazetteer, as published between 1773 and 1775, also didn't return any results to indicate house numbers, after trying out a variety of keyword searches. Unfortunately, the results are behind a paywall.
Frederick Trevor Hill's book A Story of a Street: The Narrative History of Wall Street from 1644 to 1908 contains a handful of reprinted colonial broadsides printed in New York, including this one from October 1774. It is described as "Printed by John Holt, near the Coffee-House". No mention of a street number. This would seem to confirm the search results above from Rivington's New York Gazetteer.
But, just a few years later, at the end of the war, another broadside was published in New York City, and reprinted in the same book. In the fall of 1783, George Washington triumphantly returned to New York on Evacuation Day, the day that the British withdrew their final troops from the city and the war was over. A broadside announcing Washington's arrival plans mentioned a New Yorker named "Thomas Tucker, No. 51 Water Street".
Three years later, the first New York city directory was published. Though very incomplete, the residents who are listed have their names listed with street addresses. This would continue with every subsequent city directory for the city of New York.
Brooklyn, on the other hand, still was not using house numbers as of 1796. That happened some time between then and 1823 when another city directory was published, this time with street numbers for the vast majority of residents.
In 1776, the British took occupation of Manhattan after the Battle of Brooklyn. There was a big fire known as the Great Fire of New York of 1776, which burned down about 1/3 of all the buildings in the city.
My hunch is that the street numbering in Manhattan occurred after that, under British occupation. In any case, it seems to have happened some time between 1774 and 1783, which just about coincides with the years of the war.
The street numbers were carried over after the war, though if you notice, in those earliest city directories, not everyone is yet using a street number. Plenty of people are still listed as "at the corner of Pearl Street" and so forth. So even when people did have addresses, it wouldn't have been uncommon for them to receive mail with just their name and city on it for quite some time after.
So, to answer your question, it seems that, no, New Yorkers weren't using house numbers during the colonial period. They didn't begin to use them until after 1774, and in all likelihood, not until after the Great Fire of 1776 and British occupation had begun, though there isn't any firm evidence I could find to prove that specifically. In any case, by the time the war had ended, the practice had begun and has continued since. It was some time after 1796 but before 1823 when Brooklyn did the same.