Yes I would search the death records. If you are going to focus on Tyrone then deaths in that county will have been registered in one of the following:
Armagh city, Castlederg, Clogher, Cookstown, Dungannon, Enniskillen, Gortin, Lowtherstown (sometimes listed as Irvinestown), Omagh & Strabane.
I would also check Co Antrim. The registration towns there are: Ballycastle, Ballymena, Ballymoney, Belfast, Coleraine, Larne, Lisburn & Lurgan. (Ballymena & Ballymoney are the two that cover Rasharkin)
Dunnamore does exist and it is in Tyrone. It’s in the parish of Kildress which falls within the Cookstown civil registration area. Griffiths Valuation for 1859 lists 2 Michael Monaghans there (senior & junior, so probably father & son). Both were apparently farmers though, not schoolmasters.
In the 1901 census there are about 6 Monaghan households in the townland with about 50 people of that name. I did spot 2 school teachers though:
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Ballynasollus/Dunnamore/1730521/
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Ballynasollus/Dunnamore/1730531/
Here’s one of the Michaels (he died in 1906):
http://www.census.nationalarchives.ie/pages/1901/Tyrone/Ballynasollus/Dunnamore/1730534/
Another Michael died in 1892. His death cert describes him as a farmer.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_retu…
A third Michael died in Dunnamore in 1891. Again a farmer.
https://civilrecords.irishgenealogy.ie/churchrecords/images/deaths_returns/deaths_1891/06074/4732358.pdf
John Monaghan school teacher married Mary Monaghan (same maiden name) in 1892. His father was another John Monaghan, also a school teacher.
So there were 3 Michaels in Dunnamore in the mid 1800s, but none was a school teacher, so far as I can see. What’s the source of your 1851 census extract? If I can have a look at it, I might be able to give it a bit of thought.