No arcade games (perhaps aside from some very, VERY modern ones you won't be able to afford real boards for and that MAME won't emulate) will use 1024x768. Heck, 800x600 is extremely rare, with even progressive 640x480 being unusal. The only reason you'd care about 1024x768 would be normal PC usage (desktop, web browsing, setup/config, etc.) and possibly for a MAME frontend (boo mame, gogo real boards! - sorry, I'm a hardware collector).
The Billabs has a slightly lower dot pitch than most arcade monitors, so the resulting picture will be more like what you'd expect from a big PC monitor rather than a TV or arcade monitor. This means it will look better at higher resolutions, but not as "authentic" at lower resolutions. From what I've heard, it still looks great pretty much anywhere in the spectrum.
As far as video cards, any modern video card should work. I don't know about ATi (their linux support is horrible, so I stick with nVidia), but any nVidia card from the FX5000 series or newer can go all the way down to standard arcade resolutions (480-525 lines, interlaced, 15kHz horizontal) and up to well more than that monitor is capable of. Since you don't need the plug and play standard res capabilities of the AVGA, you don't need it (incidentally, I beleive the AVGA is capable of higher resolutions, but I don't own one).
These arcade monitors are not Plug and Play in the sense that they support they DDC information for retrieving supported scanrates automatically, but you can just configure it in Windows (or Linux, or whatever) as a generic 1024x768 capable monitor and ensure that you use 60Hz. You can actually go higher refresh at lower resolutions, if you want, but using 60Hz for everything makes things easier. I think Windows has a "1025x768 LCD Flat Panel" or similar option that will force it to use 60Hz refresh. The Billabs will support the standard/default WinXP operating mode (800x600@60) as well as VGA text mode (640x480@70-72), and it shouldn't let you run any modes that will damage it.