If you like live theater productions, book yourself some tickets for a play or other stage show at The Public Theater. Check online to see what's still available.
You'll be spoiled for sightseeing choice when staying in Manhattan. The Public Theater is a good place to start here, but there are loads of other worthwhile attractions to discover. Times Square, Statue of Liberty, and Broadway are close to the arts and culture attraction and and will cap off your day nicely.