Do American universities generally sponsor H1B visas for staff? 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.