Can I buy NYC extension domain from outside using fake street address?
Sure, but I would not advocate for you doing so as it breaks their
eligibility requirements.
Even if you can incorporate in New York and have "office space" there, you still don't qualify, nor would you with a P.O. box; you must have physical residency there.
You might get lucky and never have an audit. However, it's safe to say that it might catch up with you sooner or later. The more time and money you put into the .nyc domains, the more it's going to sting when they're all seized at once.
May I ask why you want the .nyc so bad and why you can't use something else? Just trying to look out for your best interest, be it domain name investing to development.