There was a Thomas Armstrong in Falmouth in 1792, 1793, 1794, 1795, and 1817. I don't see a land petition, but that just means he bought from someone else.
You might also want to look for his father's will, as that can prove the relationship.
I've had a look around the resources of Genealogical Association of Nova Scotia, which is behind a paywall, and I don't see any church or probate records for Hants County online. There was a severe shortage of clergy in Nova Scotia, which is the reason the universities were founded around that time, but the Presbyterians were usually literate for a couple of generations at least, and I encourage you to persist.
The historical text for Falmouth has a number of references to an possibly unrelated Armstrong family, Scottish settlers who emigrated in 1777, but its possible there are also references to your family, I didn't look carefully.
The information you're looking for may be in paper or microfilm at the Nova Scotia Archives in Halifax. You may be able to arrange for a remote lookup for a fee.
Another resource to explore is the West Hants Historical Society.
