Dublin is divided by the River Liffey: North and South. Both sides are similar but the south side has the Government buildings, Trinity College, most of the National museums, and is considered to be more posh!
Beside it there's the Modern Art Gallery which is small and personally I don't like it very much, but it's free. There's also the Writer's Museum which has a little history about Irish authors / playrights, but you should pay around 8euro to enter and it's not worth the visit if you don't have much time in Dublin.
On the other side of O'Connell St. there's a statue of James Joyce.
Walk to O'Connell bridge, look to the right and you'll see the Ha'Penny bridge, which received its name from the half-penny toll people used to pay to cross the river in the past!
Walk out the Arts Block Exit of Trinity and turn left. After 100m there is the National Art Gallery of Ireland. It's got a beautiful painting of Caravaggio among many other paintings. It's beside a park called Merrion Square with a statue of Oscar Wilde. Free entrance.
Also very close is St. Stephen's Green Park, which is nice for a walk. Then go to Grafton St, another important place for shopping! Try some coffee or chocolate from Butler's Cafe! When you arrive back at the end of Grafton Street there's a statue of Molly Malone. She is famous for a song:
Where the girls are so pretty,
I first set my eyes on sweet Molly Malone,
As she wheeled her wheel-barrow,
Through streets broad and narrow,
Crying, "Cockles and mussels, alive, alive, oh!"
Walk to Dame Street and then visit Temple Bar. This area is the most cultural part of Dublin with lots of young artistic people. But it is also the most touristic part with lots of restaurants and bars. Several bars have live music and are nice to visit here for a quick pint of our national drink: Guinness!
Go back to Dame Street and walk west. You will find Dublin Castle on the left which is nice to walk around. You can take a tour for about €5. There's also a museum there with a huge collection of Oriental artifacts from China and Japan. This museum is free.
From outside Christchurch or anywhere on Dame Street, you can get bus number 123 which takes you directly back to the Marathon venue!