I search the IFHF's online database at ROOTSIRELAND.IE. It has abt 19 million B/M/D and other records for MOST Irish counties. Records before 1863 are increasingly sparse. By 1800, they are almost non-existant. Many B/M/D events either weren't recorded, or haven't survived.
For Pat(rick) Carroll B 1828 +/- 2 yrs somewhere in Ireland with the parents' names unknown, I found 128 records with 6 in Co. Offaly. One of these MAY be for your ancestor. You will need a more exact DOB, placename, and/or parents' names to pick which, if any, is the right one.
Alternatively, you can search for his death certificate. Since he died on/after 1885, this event was likely recorded by the General Register Office. Contact it at www.groireland.ie (Note - fees apply).