Can you leave it, yes, should you leave it, no. It really should have a finish coat of paint.
Yes, a roller will work just fine, a 1/2 inch nap.
A GOOD latex primer will work, Zinsser's 123, or Gardz(if you can find it)
Be sure to wait the appropriate time for the plaster to cure( if it was actual plaster). Ask the people who did the job.