The former Blyth Spartans boss takes over from Tommy Wright who announced he would be leaving last month.
Armstrong twice played for the Quakers and has since becoming a manager, he took his previous club into this season's play-offs.
But he admits the size of the challenge at Blackwell Meadows was too big to turn down.
"There's a big job to be done, and I'm really looking forward to the challenge," he told the club's website.
"I have no doubt that it will be a challenging job, but I believe that we can bring success to the club. I'm looking forward to working with everybody.
"The fans, and the ambitions of [Chief Executive] David Johnston, are big factors in my coming to the club. I want to try and get the club as high as I can."