Dublin Castle is well worth a visit; the Chester Beatty Library is superb; the State Apartments may also be open when you are there and although I haven't been there, I believe they are well worth a visit (they are used for visiting dignitaries).
Once you're finished there, head down to Trinity College, which will be pretty much right ahead of you once you leave Dublin Castle and get onto Dame Street.
For shopping, try Grafton Street and the streets around there; there are two absolutely first rate bookshops - Waterstones and Hodges Figgis right opposite each other on Dawson St (just parallel to Grafton Street and indeed, directly opposite the side entrance to/exit from Trinity.
For eating, try Fitzers in the National Gallery or there's plenty of the up along Dawson Street, such as Milano (good for Italian). Otherwise - if it's a nice day - buy a sandwich and sit in St. Stephen's Green (wrap up well!) If you want a nice cup of afternoon tea and biccies, try the newly refurbished Shelbourne. That's on Kildare Stret (which is parallel to Dawson Street) ... if you walk further down, you'll see our parliament building - Leinster House.
Other ideas; if you turn left instead of right as you walk down Dame Street, towards Trinity, you will go around by the Bank of Ireland on College Green - a very impressive sight in its own right, then up Westmoreland Street towards O'Connell Bridge and O'Connell Street. This will take you to the North Side. There, I'm afraid , I must part company ... (Lions and tigers and bears, oh my ...) Joking apart (the lions were shot, the tigers were mugged and the bears have been admitted to a stress clinic), it's not that bad. Clery's is a well known department store and Easons is a very good shop for books, music and stationery. I think there is life towards the far end of O'Connell Street, but it's only rumour and few people have got back over the Liffey to tell the tale ...