Citylink buses from Edinburgh do go to Cairnryan Stena Line Terminal (Stena Line being the ferry) Cairnryan is within the Stranraer postal district, which is maybe why it is mentioned. Bus and ferry to Belfast takes nearly 7 hours, so you'd pretty well lose a day each way as DtG said. Flights are about 45 minutes!
Beyond Belfast, the Antrim coast and glens are worth exploring: Carrickfergus Castle (currently some restoration work going on) Carrick-a-Rede rope bridge, Giant's Causeway. We spent a day on Rathlin Island: nature and bird reserves, ferries out of Ballycastle. And we had a day in Derry/Londonderry: fascinating place, if very turbulent history.