Have you found his death certificate? It might mention his parents names and/or where in Ireland he was born/from. I think 'cluster' genealogy is what might help you with this. 'Cluster' means to go out of the box and look for his siblings information to see if any of them have more than the county listed. i.e., census, marriage, death records. I hope this helps you.
Are you sure Patrick was born in Cork? My GGrandfather is John Sullivan. I was not sure if he was born in Cork or Limerick and have recently found that he was born in Limerick. His parents and Grandparents were from Cork and Tipperary. I do believe I saw a Patrick Sullivan in my list. Limerick is hard to get info from as they charge you for each record. I would be happy to see if anything solid is in my list. Please let me know.
I have looked at the Cork church records that have recently come one line. The DOB you give doesn't turn up anything much. It is not unusual for the DOB to be out; even the year of birth; I tried putting in just 1861 without much luck. The Civil registration did'nt start in Ireland until 1864 so the church records are the mainstay. These however are Baptism records; where the DOB is given in the Baptism record it could easily be out; there wasn't a lot of literacy about at the time.