(I grew up in another state, so all that local and state govmt I learned way back when as a kid is useless)
http://www.dos.state.ny.us/info/constitution.htmNo election of a lieutenant-governor shall be had in any event except at the time of electing a governor.
...
In case of vacancy in the offices of both governor and lieutenant- governor or if both of them shall be impeached, absent from the state or otherwise unable to discharge the powers and duties of the office of governor, the temporary president of the senate shall act as governor until the inability shall cease or until a governor shall be elected.
In case of vacancy in the office of lieutenant-governor alone, or if the lieutenant-governor shall be impeached, absent from the state or otherwise unable to discharge the duties of office, the temporary president of the senate shall perform all the duties of lieutenant- governor during such vacancy or inability.
If, when the duty of acting as governor devolves upon the temporary president of the senate, there be a vacancy in such office or the temporary president of the senate shall be absent from the state or otherwise unable to discharge the duties of governor, the speaker of the assembly shall act as governor during such vacancy or inability.