Hi Jan,
Thanks for your message. I hope that someone with information makes a connection with you and helps you to further your research.
Rule was a very uncommon name in Ireland during the 19th century. You can see this here:
http://www.irishtimes.com/ancestor/surname/index.cfm?fuseaction=Go.&UserID=
You can see these results in more detail in Griffith?s Valuation (this is a property valuation survey carried out in Ireland 1848-64)
http://www.askaboutireland.ie/griffith-valuation/index.xml?action=doNameSearch&Submit.x=76&Submit.y=25&Submit=Submit&familyname=rule&firstname=&baronyname=&countyname=&unionname=&parishname=
Unfortunately there are none recorded in Tyrone.
It will be difficult to find more information on the family without knowing a specific parish as most church records are still held locally. It may be an idea to contact the Irish World Heritage Centre in Tyrone for some assistance, however a fee may apply: info@irish-world.com
In the meantime here are some websites you may find useful:
The National Archives of Ireland http://www.nationalarchives.ie/genealogy1/genealogy-records/introduction/
The National Library of Ireland http://www.nli.ie/en/family-history-introduction.aspx
The National Archives UK ? genealogy search: http://www.nationalarchives.gov.uk/records/looking-for-person/
The Public Records Office of Northern Ireland http://www.proni.gov.uk/index/family_history.htm
Genealogy Links: http://www.genealogylinks.net/uk/ireland/tyrone/index.html
From Ireland: www.from-ireland.net
Co. Tyrone: http://www.cotyroneireland.com/
Family Search: www.familysearch.org
Historic Graves: www.historicgraves.ie
Kind regards,
Genealogy Support