There are 2 Hart's listed in the Sligo telephone book, both Ballymote. Address sufficient for mail delivery, no postal codes required.www.census.nationalarchives.ie
I couldn't find a listing for Michael Hart as a mason in the commercial directories.
Both Sligo and Galway belong to an Irish History Foundation with birth records before 1864. All records were Parish records until 1864 when civil registration commenced.You should be able to find a birth record for Annie Hart, father
Michael. A birth record would give you place/parish, date,
mother's name and maiden name.