I am looking for any information on my 3rd great-grandfather, James O'Hara, who was born in Altrest, Parish ofDonaghedy, County Tyrone on December 15, 1815, and any other ancestors.
You don’t say what denomination your O’Hara ancestor was but I’ll assume RC. If so, Donaghedy RC doesn’t have nay baptism or marriage records earlier than the 1860s so there’s unlikely to be any record of John or his family in church records.
I had a look for the tithe applotment records for Altrest which list farmers there in the 1820s/1830s. Unfortunately those records are also mostly missing and there’s no record of an O’Hara family there. See:
http://www.cotyroneireland.com/tithe/donagheadytithe.html
There was a John O’Hara in Ballaghalare (which is not too far from Altrest) in 1856. He might be related:
http://www.cotyroneireland.com/tithe/donagheadyratepayers.html
Griffiths Valuation for 1858 lists about 20 households in Altrest. None named O’Hara.
17 households in Altrest in 1901. None contained an O’Hara:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Dunnalong/Altrest/
On the face of it, your ancestors had left Altrest by the 1830s and given the lack of church and other records for the early 1800s there may not be record of John or his family. DNA testing might be the only way of linking up with descendants today.
He got married in the US in West Virginia in 1840. He was the first resident professional school teacher in Reedy, WV, so I assume he was at least somewhat educated.