Craig
Prepare to have the month of your life. Both courses at Portrush are great, as are Castlerock and Portstewart. Ballycastle is fun, but I imagine that you would not play it more than a couple times since it's about a 40 minute drive from Portstewart and there are only 4 links holes on the course mixed in with 6 parkland and 8 headlands holes. The views are spectacular though.
Over the border in Donegal are Ballyliffen (36 holes) and North West Golf club on the Inishowen Peninsula and a little further drive you can get to Portsalon which is one of the all time great deals. It's about 35 euro and has some spectacular linksland.
RCD is a little over 2 hours away, but you can play Ardglass while you're down there to really make the trip worthwhile.
If it were me going, I would try to play all of these courses at least once, but since you're based in Portstewart, play the majority of your golf there as well as Portrush and Castlerock. You won't need to go anywhere else after you play these 4 great courses. Do see the others since you're in the area.
ps-I can send you photos of all of these courses if you would like.