As long as you have a swap file that is large enough, most likely the outcome of a too-much-memory situation is extreme slowdown rather than outright crash. I can't really speak to exactly what will happen because we don't have any machines with less than 2gb ram to test with. So partly I don't want to claim something I have no way of verifying.
I would guess from the sound of it that he will probably be okay, although if I had to pick a point where troubles might ensue it would be the overlord throne room -- lots of giant textures in need of frequent use there.
Even so, I would guess the OS would be smart enough to switch things around so that the textures hat are needed stay out of swap. Usually that's the case. But, again, I've no way of testing that. I doubt you can get 60fps with 1gb of ram, but you may very well be able to get something in the acceptable range.
Hope that helps at least a little!