Ah, the GeForce 4 MX440. Good old AGP interface and it's final years.
You should be able to run the first gen modern shooters like Doom 3, Halo CE, Half-Life 2, BF1942, CoD 2 etc, at 30 fps and classic shooters like UT and Quake 3 Arena comfortably at or above 90 fps, just keep your resolution below 800 x 600, or drop even lower wherever you need them, but that monitor looks like it's a 640 x 480 60hz display, so you should be fine.
Try even older games like Hexen, Heretic, Blade Runner etc from the GOG Store and Deus Ex, System Shock 1 & 2, Max Payne, Serious Sam etc, from Steam assuming you'd prefer Steam over GOG and want all your library there like most people.