This is obviously a problem with the game, and not related to the age of his hardware.
I got the exact same problems, with a state of the art PC:
Windows 7 64 bit/ build 7600
Geforce 9800 GX2, driver version 190.40
8GB RAM
Intel Core 2 Quad CPU Q9300
Graphics is all messed up, missing textures, trees are shown as bunch of squares. Framerate is smooth though.
I play a lot of high end games on my computer, on the highest settings, and ARMA2 is the only one that has any trouble, so its clear this is caused by bugs in the game. Very disappointing :(