PACER costs: An access fee of $0.10 per page or $2.40 per document with an audio attachment.
IDK if the court clerk will or can do what you are requesting. Personally I wouldn’t bother unless very important.
NY Supreme Court, NY county is the state trial level court in nyc. Kings county in brooklyn. SDNY federal court in nyc. EDNY in brooklyn. Fed courts all use pacer but may have different protocols as to how they update their files. Nearly all this info and more is available online if you look. Start with the court website. SCROLL is the acronym i believe (al for ny county). There’s also something called webcivil and webcrims for many state counties in ny. Good luck.