Hi,
I live in Madird, and am just back from a nice outdoor lunch in the splitting sun!
There's loads to do and see - depends what types of things you like.
There are the 3 art galleries - Prado, Reina Sofia and Thyssen. They are all within short walk of each other.
The Retiro park is nice to walk around for a few hours.
You can do tours around the Bernabeu stadium.
A good way to start is to do one of the open-top bus tours, since you get a city in perspective. They go from Puerta del Sol (big square in the center), and cost 13E for a day ticket I think (copany is called Madrid Vision).
Then the all the different areas to walk around and get to know - the older disctricts of Huertas and La Latina with windy streets to the upmarket shops of the Salamanca district.
Most people rave about spanish food - its also very cheap in comparison to Dublin - a three course lunch with a glass of wine will typically cost about 10E.
Madrid is also known for its nightlife - things start really late - restaurants open about 9:30pm, but no-one goes till about 10:30, then bars hot up about 1:00am, and clubs go all night..
Its a bit noisy, so if I were you I'd request a hotel room at the back of your hotel.
Enjoy it!