Short answer in case you want to skip the rest: Connect the new monitor, it's going to work fine.

If the computer is 6 years old it's most definitely AGP for the video card. If it were 10 then I'd believe PCI. 256MB video cards were also most definitely AGP-class cards.

I'd be very surprised if that card can't do 1920x1080, since ATI cards of the time could easily handle 1920x1200 to an LCD.

I'm quite confident that your card will handle the new monitor's resolution. The driver will definitely not show resolutions your current monitor does not support if it properly recognizes your monitor and its specs (via EDID).

But I'm also surprised that a current 27" monitor is only 1920x1080 and not 2560x1440. That card will not likely support 2560x1440 unless it supports dual-link DVI.

Connect the new monitor and you should be fine. You might want to just make sure you're running up to date drivers.