Was New York City One of the Original Thirteen Colonies?

No. New York City was not a colony—it was a city within the Colony of New York.

The Thirteen Colonies were British colonial provinces that declared independence in 1776:

1. New Hampshire
2. Massachusetts Bay
3. Rhode Island
4. Connecticut
5. New York
6. New Jersey
7. Pennsylvania
8. Delaware
9. Maryland
10. Virginia
11. North Carolina
12. South Carolina
13. Georgia

New York City (then New Amsterdam → New York) was the principal port and largest settlement in the Province of New York. The colony's territory extended far beyond the city—into the Hudson Valley, Mohawk Valley, and westward to the Great Lakes.