Honestly the gtx 260 shouldn't factor in too heavily to this situation. A 24" monitor is a pretty big step up from a 22", and if you want it and you got the cash I'd go with 24" regardless. Monitors can last a long time whereas videocards are old after a year and retired to the cpu graveyard after 2-3.
Whether the 260 will play well with 1920x1200 depends on the game and graphics settings. Generally a 260 is more than enough power for anything at 1920x1200 with pretty high settings, but if you crank up all games to the absolute max you'll need a 295 for that. To put your mind at ease, my old cpu uses a 8800gt and ran 1920x1200 with Crysis, Gears of War, Fallout3, Ut3, Supcom, Red Alert 3, Bioshock, and more. I use high settings for all of them and each ran smooth as butter except crysis, which was playable at 1680x1050 on custom medium/high settings. A 260 has a lot more horsepower than a 8800gt. Also my friend uses a 260 and a 24" LG panel, games on it all the time w/o problems.