There are no Narey spelled this way in the current Sligo
1833 Mathew Narey Parish Kilglass, Townland Carrowhobbok Orme
1833 Patrick Narey, Parish Skreen, TL Leekfield
Next Census substitute is 1857 in Sligo the Griffith Valuation, there are 23 properties in the name of Andrew, 1 Parish Achonry and 2 in Parish Kilmateigue (which is probably 1 with 2 properties)
5 John Narey's in Parish's, Kilmacshalgan or Kilmacteiguewww.askaboutireland.iewww.census.nationalarchives.ie
Sligo does belong to an Irish History Foundation with birth and marriages prior to 1864, you have an approximate year so you should be able to find a birth record depending on the survival of Parish records. I just took a look at the above parish's and neither has parish records before 1877.
The other way of possibly finding connections would be
gravestone inscriptions, many county Heritage societies have done survey's of graveyards. Also any deaths after
1864 you can find on:www.familysearch.org
Andrew Neary death registered Jan-March 1889 Ballina, Sligo
aged 77, est year of birth 1811 based on age given at death.