You don’t have to go far to find rather a lot of web pages about the supposed connection between Armstrong and Fairbairn.
From the Armstrong DNA project at FamilyTreeDNA:
“Another tradition, frequently mentioned in Scottish clan surname guides, concerns Fairbairn, an armour bearer to a king of Scotland. He is claimed to have rescued his master when he had his horse killed under him in battle. Fairbairn allegedly lifted the unseated king onto another horse using just one arm. However, the weight of medieval armour makes such a feat appear highly unlikely! ”
If the Fairbairn chap with the strong arm was from the Scottish Borders, yDNA testing (direct paternal line) in the Fairbairn Surname DNA project is showing us that the majority of the tested Fairbairn lines from there are haplogoup I1, and in particular, a sub branch (I1-Y7277) formed about 3000 years ago.
The Armstrong DNA project introduction states “Y-DNA results show that the mainstream Armstrong descent cluster belong to the R1b-312** Haplogroup and are part of the S389/L624 and R-Z30600 subtypes.”
Which makes it a no connection between the two, or at least between the majority groups in both DNA projects.