Well first off all not all 24" monitors are at 1920x1200, about half of the models are 1920x1080. Secondly you do
want to run your monitor at native res where ever you can so if your GFX card is going to struggle at 1920x1200 or 1920x1080 then you will simply need to upgrade the GFX card or get a lower res monitor. Lastly 1920x1200 or 1920x1080 makes a big difference to games, especially FPS games, TBH the higher the resolution the bigger advantage you have.
So, if you want to run new games at a high setting but dont want to buy a new gfx card then get a lower res 22", otherwise, the more pixels the better