If you absolutely have to fly into Dublin, and since you only have a week, I would actually skip Dublin at the beginning and head west when you arrive. To go from the Dublin airport to, say, Connemara National Park, takes about three hours and forty-five minutes. To me, this makes more sense:
9/9 Arrive Dublin, pick up rental, drive to Galway..stopping at Trim Castle, Athlone Castle & Visitors Center and Clonmacnoise. Stay in Galway.
This would set you up perfectly for visiting the Connemara area on 9/10
9/10 Connemarra area (Ross Errily Friary, Ashford Castle, Cong, Kylemore Abbey, Connemara National Park, Clifden Castle, Aughnanure Castle…to name a few). Stay in Galway a second night.
9/11 Drive Galway to Bothar nA hAillite in the Burren, Doolin (Doonagore Castle), Cliffs of Moher, Kilkee Cliffs, take the ferry from Killimer to Tarbert, stay in Listowel.
This would put you less than an hour’s drive to the Dingle Peninsula.
9/12 Drive from Listowel to the Dingle Peninsula…going through Conor Pass, Dingle, Slea Head Drive, loop around back into Dingle for lunch, Minard Castle, Inch Beach, Gap of Dunloe, stay in Killarney.
9/13 Drive Killarney to the Ring of Kerry….stopping at Ballycarbery Castle, Valentia Island, Kerry Cliffs, Waterville, take the Ballaghisheen Pass to the Ballaghbeama Gap, Molls Gap, Killarney National Park, stay a second night in Killarney.
9/14 Drive Killarney to Blarney Castle, The Vee (Graigue, Co. Tipperary), Rock of Cashel, stay in Kilkenny.
At that point you’re only 90 minutes from the Dublin airport.
9/15 Visit Kilkenny Castle, then drive to your hotel near the Dublin Airport (I suggest Howth)….stopping at Glendalough, Glenmacnass Waterfall, Lough Tay, Sally Gap. Stay near airport.
9/16 Return car, depart.
Yeah, it's a busy itinerary, but you'd get to see a lot of Ireland in a short amount of time. Also, you may or may not have heard of some of the places I've mentioned. Just go to Google images to see if they interest you.