Do top American universities (such as NYU, etc.) sponsor H1B visa petitions for staff position? If yes, what type of jobs are sponsored in staff positions?
Yes. A simple google search for "H1-B university staff" links to a bunch of federal and school-specific policies. It goes without saying these roles are reserved for people with "specialized knowledge," etc. Also a link to a similarish question about faculty is here.
You asked "what type" and I missed that the first time around. Here is a link to a page that explains H1B jobs open at NYU. Not all but mostly faculty positions.