There would seem to be two options for this trip. Both involve a couple of days seeing in Dublin and then going to another part of the country.
The first is Dublin and the North. Spend a couple of days in Dublin to see the sights, the golf courses you can see without venturing too far from the city center are Portmarnock, The Island and Royal Dublin. After Dublin drive up the east coast to Newcastle, Belfast and Portrush.
It is possible to see County Louth(Baltray) on this leg of the journey. In Newcastle is RCD, should have no problem gatting a tee time that time of the year, plenty of great scenery for your better half as well, it really is a great walk. Then on to Belfast to see amongst other things the Murals from the troubles. There are Mural Tours, really are amazing to see. And on to Portrush from there, see Royal Portrush but also the Giants Causeway, Carrick-a-Rede Rope Bridge and the whiskey distillery at Bushmills. A great mixture of golf and culture on this leg.
The other option is Dublin and the West. Spend a couple of days in Dublin again and drive to the west coast, see Galway, Connemara, maybe a ferry to the Aran Islands, drive down the coast to Clare and the Burren, traditional music in a pub in Doolin on the way, onto the Cliffs of Moher and Lahinch. Drop into Lahinch golf club and further down the road to Doonbeg. This should take about four days as well. I also would look at B+Bs to stay in, won't be busy that time of year and would be open to negotiation, much better then hostels, the family of the house would let you in on hidden local secrets as well.
This is what I would recommend. If you have any questions, ask.