You shouldn't really worry about using a non-metallic screwdriver as the brass screw of the pot should be electrically isolated from the circuit.
There's no high frequency RF circuits in that region which will be effected by a metallic tuning tool either.
They probably suppled a plastic tool becuase they're cheaper, and to minimise electrostatic discharge risks inside the computer when you go poking a metal screwdriver in there.
@Crusher
No idea what you mean by "coalface". Maybe carbon??
The modern multi turn pots are usually made of stuff called cermet which is kind of like conductive ceramic.
As for tuning it, normally you find a "peak" in the tuning range.
So you keep turning it one way until you start to see the image getting better. Now keep turning while counting the number of turns until it starts to degrade, then go back by approx half the number of turns you counted to put you roughly in the center of the tuning range.
If the picture remains crappy, make sure the monitor cable is securely plugged in.