I imagine you're somewhat interested in history and that sort of stuff because you play this game?
Never been there, but . . .
I'd go to the National Museum of Ireland. George Bernard Shaw, the Fabian socialist and playwright, who never received a formal education, said he learned everything he knew from this museum, and when he died in the 1950's, he left his entire estate to it.
All that money from Pygmalion really helped the museum a lot, and they've been able to make a lot of acquisitions and expand the museum.
You might also want to vist the famous downtown Dublin Post Office where the Easter Rebellion occurred in 1916. It was a disaster, and its leaders were executed by the British; but was a political and social victory, and helped paved the way to Irish Free State Independence.
On a more fun note, there's one pub called Davey Byrne's or Danny Byrne's where a lot of Irish writers like Brendan Behan (who had his "office" there) and I believe James Joyce hung out at. Sounds like a good place to have some Guinness (sorry, I don't know where it is located, but I saw it in a recent book about Irish pubs at the bookstore).
Certainly (if you're real high brow) go to the Irish Parliament, National Library (I'm sure they got one), and Trinity College (the traditional Protestant University, thought now it doesn't discriminate based on race), and Queen's University (set up in the 19th century as a non-denominational university by Queen Victoria). If they have a Stock Market, that could be interesting too.
So, how about posting a few pictures of the sites you saw when you get back?