You should give your employer your new address now. There is no point in waiting. It's too late to change the withholding for this year, but at least they will start withholding NYC tax next year. Also, if they are going to send W-2s by mail, ask them to be sure to mail your W-2 to your current address.
NYC tax is included on your NY state tax return. The NYC tax that you owe for 2022 will be calculated on your tax return and included in the total tax. So yes, you will pay the NYC tax when you file your 2022 New York tax return.
