Definitely turn in your rental car before Dublin. The parking is expensive and the traffic is horrendous. You will do quite well with the ground transportation.
Welcome to the forum. I will chime in with everyone else's advice. When staying in Dublin city center a car is not advised. Only if you were staying in the burbs like Malahide would I suggest keeping the car. But even then you would take public transportation into the city. You can get around fine by walking, bus and DART. Go to my "Transport" page to click through to the various links.