Do I need to file New York State income tax if the investment property net income is zero.

Can you clarify if the real estate property was sold?

The property is not sold.  It is a long-term rental property. After expenses and depreciation, the net income is ~$1000.

Yes, you should file the New York tax return, especially if you did not make a profit.  The loss will be carried forward in the state for future use.  If you don't file, you won't get credit for it.

I made a small profit in this case ($1000).

If you have a rental property located in the state of New York, you need to file a New York tax return whether the property had a profit, loss, or broke even.