If nobody in wider genealogical community can provide you with new information as to the origins of your Armstrong ancestors your next step will be to conduct your own research based on the information you hold from the marriage certificate, namely:
Joseph Armstrong, age 30 of Woodhill, son of John Armstrong
married on 15 June 1861
Jane Spratt, age 22, of Coolbuck, daughter of William Spratt
in Irvinestown Presbyterian Church in Derryvullan parish, County Fermanagh
As births in Ireland were not subject to civil registration until 1864 it means you will have to rely on church baptismal registers to confirm the birth details of Joseph Armstrong, son of John, born c. 1831 and of Jane Spratt, daughter of William,born c. 1839. Dates of commencement and quality of information in church registers vary from parish to parish and from denomination to denomination. Access to church registers, in the absence of indexes and databases, is generally gained through knowledge of an ancestor's parish address and religious denomination. There is no national index to Irish church registers. To date, only the county-based genealogy centres have attempted any large scale, systematic indexing of church registers in their localities.
Although RootsIreland, at www.rootsireland.ie, is the largest online source of Irish church register transcripts, it must be emphasised that a failure to find relevant birth/marriage entries in this database doesn't mean that the events you are looking for didn't happen in Ireland. It simply means that they are not recorded in the database; for example, they may be recorded in a record source which doesn't survive for the time period of interest or in a source that has not been computerised.
If you wish to identify all available church registers in the wider Irvinestown area, microfilm copies of which are held in our national archives in the Public Record Office of Northern Ireland (PRONI) in Belfast you should explore their Guide To Church Records, which can be accessed on their website at https://www.nidirect.gov.uk/publications/proni-guide-church-records
as it lists, in alphabetical order by civil parish, church registers of all denominations for most parishes in Ulster and their commencement dates, together with their microfilm reference details. You can examine any church registers held in PRONI at no charge, if you visit this office. If you require a researcher to examine these registers on your behalf you would have to pay for their services.
The problem you will face, and it is one that all people tracing their Irish ancestry face, is that once your research extends beyond church registers (which are the building blocks of Irish family history) you have few means (unless the family history has been documented and passed down through the generations) to confirm family connections.
Quite often the only realistic strategy in tracing ancestors beyond church registers is to examine surviving land records and census substitutes, often compiled by civil parish, for any references to a surname or given name of interest.
There are a number of census substitutes (e.g.1630 Muster Roll, 1663 Hearth Money Rolls, 1740 Protestant Householders Lists,1766 Religious Census, 1796 Flax Growers Lists, early-19th century Tithe Books and mid-19th century Griffith's Valuation) which can be searched to confirm the presence of the family name. The problem with these sources is that they name heads of household only; hence they provide insufficient information to confirm the nature of linkages between named people in these sources. Census substitutes, however, are very useful in confirming the presence of a family name in a particular townland and/or parish, and in providing some insight in to the frequency and distribution of surnames.
Griffith's Valuation can be examined at no charge at www.askaboutireland.ie/griffith-valuation. You can search, for free, a number of 18th century census substitutes for Northern Ireland, such as indexes to pre-1858 wills, 1740 Protestant Householders Lists and Religious Census of 1766 in ‘Name Search’archive on website of the Public Record Office of Northern Ireland at https://www.nidirect.gov.uk/information-and-services/search-archives-online/name-search.
At www.ancestryireland.com/scotsinulster you can also search, free of charge, by surname, the Flax Growers Lists of 1796, the Protestant Householders Lists of 1740, the Hearth Money Rolls of the 1660s and early-17th century Muster Rolls for Ulster. Although such sources will confirm the presence of a surname of interest they will not confirm if there is a connection between people with the same surname!
It is quite likely that your Armstrong ancestors in Ireland trace their ancestry back to Scotland. It is estimated that 100,000 Scottish migrants settled in northern half of Ireland during the 17th century Plantation of Ulster.
By the end of the 17th century a self-sustaining settlement of English and Scottish colonists had established itself in Ulster. One estimate of British population of Ulster is 40,000 by 1640 (with 60% of Scottish origin), 120,000 by 1670 and 270,000 by 1712. It is also estimated that by 1715, when Scottish migration to Ulster had virtually stopped, the Presbyterian population of Ulster, i.e. of essentially Scottish origin, stood at 200,000.
An excellent overview of Scottish contribution to the 17th century Plantation of Ulster will be found in the following book: 'The Scottish Migration to Ulster in the Reign of James I'by M. Perceval-Maxwell (published by Ulster Historical Foundation, Belfast).
Armstrong is among the fifty most common Ulster surnames, and the third most numerous in County Fermanagh. This well-known name from the Scottish Borders came with numerous Scottish immigrants at the time of the Plantation of Ulster in the 17th century. They settled mainly in Counties Fermanagh, Tyrone and Cavan.
The Armstrongs were a border clan who trace their descent from an armour bearer to a king of Scots who rescued his monarch in the midst of battle when his horse was killed under him. From this deed, the family came to be known as ‘Armstrong’ and received a gift of lands in Liddesdale in the western and middle Marches of England and Scotland. The valley of the Liddel Water which rises in Scotland, close to the border with England, flows south-westwards to the border and joins the Esk before entering the Solway Firth, was home to the Armstrongs. The Armstrong clan became very powerful, and at the height of their power in the 16th century they could muster an army of 3,000 men.
From the 14th to late-17th century, the border between England and Scotland – the Debatable Lands – was a turbulent place. The Border country was ravaged by the lawless Reiver families who stole each other’s cattle and possessions. They raided in large numbers on horseback. This type of life resulted in the growth of large closely-knit family groups with intense clan loyalties and fierce feuds against others.
Prior to the Union of the Crowns of England and Scotland in 1603 the Scottish Border was divided into three districts; the east, west and middle Marches. Each March was presided over by a warden who settled disputes with the warden of the appropriate March in England, as border warfare was rife at this time with frequent cattle raids.
The Armstrongs were the most feared and most dangerous of the reiving families, and one of the most infamous was William Armstrong of Kinmont, known as Kinmont Willie. Unlike other reivers he liked to ride by day rather than under cover of darkness. The year 1593 saw his biggest raid. He rode with an army of 1,000 men who stole 2,000 cattle and £300 worth of goods. Three years later he was captured and imprisoned in Carlisle Castle, only to be set free by his sons in a daring rescue.
When the power of the riding clans was broken by James I in the decade after 1603 many came to Ulster to escape persecution. This flight to Ulster also suited the needs of the king. James I, from 1610, was determined to implement a deliberate plantation of Scottish and English colonists on the forfeited estates of the Gaelic chiefs in Counties Armagh,Cavan, Donegal, Fermanagh, Londonderry (then known as Coleraine) and Tyrone. The death of the 10th chief of clan Armstrong in 1610 also encouraged clan members to scatter.
The Armstrongs settled particularly in County Fermanagh. These Border families of Scotland were well suited to life in the frontier of the Plantation of Ulster.They were a resilient people who stayed in County Fermanagh throughout the upheavals of the 17th century. Scottish settlers were hardier than their English counterparts, and the Borderers were even better adapted again to life on a new, insecure frontier.