I search the IFHF's online database @ ROOTSIRELAND.IE. It has abt 20 M church/civil/census B/M/D records for MOST Irish counties, including Donegal. Records before 1863 are increasingly sparse - almost non-existant by 1800. Protestant records are the LEAST likely to have survived.
For James Harrison B ????, Co. Monaghan, with the parents' names unknown, I found 3 records. Will leave it to you to visit the database and pursue this further.