Search the Irish Family History online database at ROOTSIRELAND.IE. It has abt 20M B/M/D records for most Irish counties. Records before 1863 are increasingly sparse: almost non-existant before 1800.
Since you have one marriage after 1863, this should have been recorded through mandatory civil registration. Civil records SHOULD be in the above database. If not, then you should contact the General Registration Office in Ireland.