You need to do several things.
1) Get your computer to output CGA timed video. Most modern video cards are capable of this, but they need a little coaxing since most people have no desire to run that mode, and hence there's normally no easily selectable option. In Windows, try PowerStrip or the "Soft 15kHz" thingie. In Linux, it's a simple matter of chucking the proper modeline in your xorg.conf. Any nVidia card GeForce4 TI or newer (so any of the 5/6/7/8000 series, too) should work. ATi, I seem to recall anything that's newer than the now rather ancient 9000 series works.
2) Get your sync signals correct. If your monitor accepts separate sync, use that. Most PCs by convention output positive separate sync, so be sure to set things up correctly. Most video cards do support composite sync, again with the caveat that it's not always easy to configure. You can try that, too. In this case, you might as well set things up for the arcade convention of negative composite sync as the polarity adjustment is usually right next to the sync type adjustment. I could tell you exactly how to do this in Linux, but I have no clue about Windows. Some video card drivers do provide a GUI for this, though. I know my old ATi Rage Pro did.
By convention in the PC world, composite sync appears on the horizontal sync output, with the vertical output unused. Despite what that diagram recommends of twisting two sync wires together, I cannot recommend it. I will often work since CMOS/TTL low-side drivers tend to out-drive their high-side drivers, but it's not exactly a good thing. Far better to just get hte device to output real composite sync in the first place. If there's no option, you can build a very simple circuit (one readily available chip) to do it for you, including polarity fixing.
Note that sync-on-green, sometimes used with professional workstations, is not readily compatible with arcade monitors. Fortunately for you, PCs almost never use this without being specifically told to.
The ArcadeVGA is basically a bog-standard ATi card with a bunch of software and some bootup mods to give you a pretty interface for all this. There's no reason that this device is REQUIRED, despite what many may tell you. For me, I see no reason to spend that kind of money, but if you don't want to futz with it, it may help you out. From what I've seen, the device (and mostly the software/drivers it ships with) certainly do make things a bit easier. If you can't get things to work with your current setup, you may want to try this.