The employee spent six days in hospital with three cracked ribs and bruised lungs after becoming trapped in a rotating, unguarded shaft between a tractor and a water bowser last May.
The Northumberland-based company was fined £3,000 and ordered to pay £1,756 in costs at the Bishop Auckland Magistrates' court hearing. Mr Armstrong was fined £1,200 and ordered to pay £500 costs.
The incident occurred at Bishop Middleham Quarry near Ferryhill, Durham.
Health and Safety Executive inspector Martin Baillie said a guard on the machine had become damaged and was removed by Mr Armstrong, who instructed the worker to operate the tractor and bowser without the guard in place.
He said: "It later transpired that the risk assessment carried out in 2004 for the tractor and bowser required checks on the guarding to be carried out on a daily basis and for these checks to be recorded. However, at the time of the incident these checks were not being carried out.
"The HSE does not take the decision to prosecute individuals lightly. However enforcement action will always be considered where a person at work, particularly one in a position of responsibility, has by their actions endangered the safety of work colleagues."