You don't say where in America your ancestor John Arthur/McArthur came to from Ireland in 1811 (Canada or the U.S.?). Certainly makes a difference in the records you might find. But since he was an adult and born abt 1790, you might be able to identify WHERE he may have been born and lived in Co. Tyrone. Then you can search for church records and other records in that particular parish, village and/or townland.
There are several types of records that can show you where Arthurs were living c. 1790-early 1800s in Co. Tyrone. The Flax Lists, Freeholders records and Tithe Applotments are the most important.
For a good basic index to the Flax List (1796), see: www.failteromhat.com
Which shows you that these Arthurs were growing flax (I'm not searching McArthurs for you, just doing plain "Arthur" as an example):
Matthew Arthur, John Arthur, David Arthur - Cappagh parish William Arthur, John Arthur, Robert Arthur - Ardstraw parish Mary Arthur - Drumragh parish Alexander Arthurs - Clogher parish Robert Arther - Errigal Keerogue parish
I copied a few other ARTHUR records when looking for an illegible "A" surname of a Co. Tyrone bride. Freeholders records, 1795-1796, show ARTHURS in Donaghmore parish, and there were probably Arthurs in other parishes (I was looking specifically for Donaghmore and surrounding area). You can view original surviving Freeholders records on the PRONI web site, www.proni.gov.uk. Note that not all parishes have surviving freeholders' records.
Tithe Applotments of 1820s-1830s also show Arthurs in Donaghmore parish, as well as in Pomeroy parish. Note that Ancestry's index of the Tithes (for those who use Ancestry indexes and databases) does NOT include all the correct entries! If you type in a search for Arthurs in Tyrone Tithes, you'll get nothing. If you type in a search for everyone in Co. Tyrone in the Tithes, you get no Arthurs. So, if you ONLY go by Ancestry's index to the Tithes, you may think your ancestors weren't in Co. Tyrone.
More complete tithe lists, transcribed by volunteers from the original tithe pages, are on this excellent Co. Tyrone web site: