Does a "6-day itinerary" mean six full days on the ground, not counting the day you land and the day you fly out? That would be seven nights in Ireland. Or are you counting your arrival and departure days, with only five nights on the ground?
Here are our host's recommendations for varying amounts of time in Ireland. Without a car your choices are more constrained. I'd suggest focusing on Dublin and the west coast, based in Galway. Your plan gives you only one full day in Dublin, which isn't much. I'd add at least another day there. Galway connects easily to Dublin by train or bus (coach), and can be a base for guided tours to the Cliffs of Moher/Burren, the Aran Islands, or Connemara. Galway also connects directly to Dublin Airport via Aircoach, so you don't have to go through Dublin to get to Galway.
If your flights aren't set yet, you might save time by flying into Dublin and out of Shannon, or vice versa. How doable that is also depends on whether there are reasonable choices for such an "open jaw" itinerary from your home airport. Saving some time in Ireland may not be worth spending lots of time with multiple flights.
If you're looking at just five nights on the ground, it might be wiser to focus on Dublin and take some day tours to nearby places like Newgrange and/or Glendalough. You'd miss the scenic west coast but you'd still see countryside and some very interesting places without having to change hotels.