I would recommend that you obtain your great grandfather's marriage record which should have his age and perhaps his father's name on it. This will give you a better idea of the year when he may have been born. The website Antoine highlighted in the previous message has marriage records for the Republic of Ireland up to 1945. If you know when he died, I would also recommend obtaining his death record as this will also contain his age at death (although death records are not always 100% accurate in terms of ages as it depends on the informant's knowledge of the deceased's age). This website www.irishgenealogy.ie also has death records up to 1970.
Please note that the father's name on the birth record of an illegitimate child could not be included unless the father himself was present at the registering of the birth to prevent women from naming individuals who were not the actual father. The child therefore takes the surname of the mother. I have seen some illegitimate children being given their father's surname as a middle name.
Online birth records are only available at the moment up to 1920.
I would recommend you contact Cavan Genealogy - Email: email@example.com
Best wishes for your search,
Ulster Historical Foundation