IANAL, but as a tenant you have rights even without a written lease. Your landlord has a right to access the property but you have a right to a reasonable notice period, which is typically 24 hours. I do not know New York's laws but you can likely find information about this online. But if it's 24 hour notice period, that would mean the landlord can show your unit if he informs you 24 hours prior that he plans to do so.

Despite not having a written lease you are also typically entitled to a written termination with a reasonable notice period to vacate the property. I do not know NY's laws but where I live if you are not on a written lease you are automatically on a month-to-month lease as a tenant and there are legal protections for you in place and minimum requirements that every landlord must follow.