When your employer makes a substantial change to your employment arrangement, then whether you accept or not, makes a big deal with your UI. NY has a terrible partial UI formula. Work a day, even as little as 5 minutes, and that is a day of work, and you lose 25% of your UI check. Your employper is playing games. You were FIRED from your full-time job, and you may or may not chose to accept the new part-time, on-call position. So long as you don't work one day at the reduced hours, then you'd almost certainly be eligible, but once you agree to the reduced hours, then that becomes your job. Once you accept a job, you then don't just get to quit and except to get UI benefits.
NY will make a determination as to whether a part-time, on-call postion is suitable so the more info you have about what they are offering the better. NY will consider whether the conditions of work are prevailing so if others that do that type of work are tradtionally full-time, then a part-time, on-call offer would be less than prevailing and unsuitable. NY will also consider the time, distance, and cost of the commute, and the amount received in remuneration and if it's out of line or not worth it for what others doing similiar work get, then the job can again be ruled unsuitable.

