How long will a Radeon HD 5570 last on 1280 x1024

It is pretty mediocre already to be honest. MW2 isn't a particularly intensive game. The card is still fine for that resolution but when you try games that push it harder, perhaps BF:BC2 or Far Cry 2, you are going to have to turn AA off or even some other settings down slightly for things to be smooth. The most intensive games like Crysis/Metro 2033/Stalker will give it even more problems.
That said I doubt games in general are going to get that much more intensive until there are new consoles so it may last you a while if you are ok with it's current performance.