Probably not a good idea to paint or do anything to the ceiling right away. My advice is to simply wipe clean the mold that is showing and keep an eye on it for reappearance. You can wipe it clean by using a mixture of 1 part household bleach and 3 parts water. Preferably using a spray bottle. Let the solution stay on the surface a couple of minutes before wiping away.
Now, let's talk about where the moisture is and the builder's suggestion. The space between the ceiling bead board and the roof is highly likely to be holding moisture or dampness. Perfect conditions for mold to exist. This is the reason I suggest to first use a "band aid" approach and see if the area above the ceiling will dry out on it's own. I would involve the builder (contractor) in this discussion and let him know that although you are willing to try and let this resolve itself you do feel the problem is his responsibility if this doesn't work out. His suggestion of using "Kilz" as a primer is good but not this early. Kilz will definitely seal in whatever dampness may be there. Further, Consumer Reports Magazine has reported that mildecide in paint is of now particular value. If the mold does continue to reappear discuss with the contractor the idea of removing some of the ceiling rather than the roof itself. The problem/responsibility is his and all you are doing is offering a less costly approach to correcting the problem.
Dennis (cut2it)
C&onstructi&on C&onsultant
Over 40 yrs. as General Contractor for new const. & remodel - retired with time for you.