We are taking the fast train from Dublin to Belfast on Monday morning, August 15. Is it recommended that we buy tickets in advance or should we buy tickets at the station?
Buy a webfare from IR - http://www.irishrail.ie/fares-and-tickets/dublin-belfast
Note that the train is not particularly fast.and is more or less on par with travelling by coach, if the departure time doesn't suit you..
Irish train tickets are cheaper online than in the station, even if you don't buy them very far ahead.