Ceiling sheetrock for basement apartment
Yes, 1/2” thick gypsum board can be installed on 16” o.c. Joists for ceilings with batt insulation. (24” o.c. spacing would require 5/8” thick gypsum board without sagging between supports.)

The Code does not require any special fire separation between units, because there are only 2 units. I also doubt a Building Permit is required because it’s not structural, fire code, energy, or exit related, etc. (That’s why we don’t get a Permit for replacing carpet, painting, etc.)