Ignore 0 2 b a rebel mulligans is a lovely pub.
In the city I would check out:
The Imperial Hotel on the South Mall is a very nice “old world” hotel in the city centre - http://www.flynnhotels.com/Imperial_Hotel/
- St Finbar’s Cathedral - http://www.buildingsofireland.ie/niah/search.jsp?type=record&county=CO®no=20503418
- University College Cork - http://www.ucc.ie/en/VisitorstoUCC/
- The English Market - http://www.corkenglishmarket.ie/
- The Franciscan Well microbrewery - http://www.franciscanwellbrewery.com/
- Cork City Gaol - http://www.corkcitygaol.com/
I would go to a town called Cobh – the port where many emigrants left Ireland for the Americas - http://www.cobhheritage.com/index2.html. Cobh is approx 30 mins drive from City Centre. There is also a commuter train serving the town.
The Jameson Heritage Centre in Midleton in East Cork is worth a visit if you like your whiskey - http://www.jamesonwhiskey.com/Heritage_Midleton_experience_Tour_info.aspx
The Beara peninsula in West Cork is fabulous. The Mizen peninsula is also worth a visit with unique villages such as Glandore, Union Hall, Schull, Ballydehob. Skibbereen town is also worth a visit.
If you are interested in Irish history, when in Clonakilty you should check out the Michael Collins Centre - http://www.michaelcollinscentre.com/.