Unlike in the modern period, in Ireland in the eighteenth century elections to parliament were not held regularly. Irish parliaments could last for the life of the sovereign (e.g. George I and George II), until 1768 when the Octennial Act ensured an election every eight years. The death of an MP, and before the 1790s nothing else, created a by–election.
The Act of Union allowed for 100 Irish MPs: the counties and Dublin city retained both members but Dublin university and the larger towns were reduced to single members allocated by agreement or lot. There was no general election immediately following the union and the surviving MPs were simply transferred to Westminster where as British MPs they have biographical enteries in the British History of Parliament. Only county MPs are listed here, borough MPs are given in the list of MPs.
The election results database for the Irish Parliament is a simple way to see who was elected for which constituency in any given parliament. You can then follow the links on the results page to go view the biography of an MP or to a history of a constituency, at a cost of £5.00 per biography, or £3.00 per entry for County/County Borough, and £1.00 per entry for other Constituencies.