I'm not admitted to practice law in New York, so you need to ask a New York lawyer. But if this were California, you would absolutely be able to file a claim for funeral costs. They would be a "priority claim" that would have to be paid after taxes and before lawyer's fees and executor's fees.
In California you only have a limited time in which to file a claim in the probate proceeding (4 months after "letters of administration" or "letters testamentary" are issued). Other states have different rules, so make sure you file your claim on a "timely" basis or it will be barred.
The information you obtain at this site is not, nor is it intended to be, legal advice. You should consult an attorney for individual advice regarding your own situation.
A funeral creditor (you) has priority over all other claims and expenses of the estate. The law requires that you be reimbursed from the first monies coming into the estate.