David,
Regarding Doonbeg, I will 2nd the opinion of those who think Lahinch, Ballybunion, and Tralee are more important visits. Lahinch is only about 35 minutes from Shannon Airport. It is a must play, when I'm in the area I always set up for two rounds there. It would be a great introduction to links golf.
Ballybunion is another course that merits playing twice, certainly one of the best links on the planet. 45 minutes south is Tralee, which features one of the worlds more amazing back nines, and some of the most gorgeous dunes in all of Ireland. All three of these courses are substantially superior to Doonbeg, in my opinion. Doonbeg has great lodging and the course has a number of very good holes, but there are several that are architecturally poorly laid out and the course falls behind Lahinch, Ballybunion, and Tralee for that reason.
I don't know your itinerary, but if you could get off the plane and play Lahinch, stay locally,play it again, then head south towards Ballybunion you might be able to get an afternoon tee time that would avoid members time. The trip between the two courses is 2:00+ hours depending on how long you have to wait for the ferry at Tarbert.