Some PC video cards (I have no clue about the ArcadeVGA) can actually output composite sync on their own. It would be on the horizontal sync (VGA pin 13) line which you can then connect directly to the monitor. Those cards will also usually let you control polarity to get active-low (negative) sync.
If you can't get composite sync directly from your video card, the correct way to do it is XOR active high vsync with active low hsync. You can do this with a 74[LS/S/CT/HCT/VHCT]86 chip (including using an extra gate to invert hsync since VESA says they should both be active high for PC video). In reality, you should never connect the two sync OUTPUTS together (connecting a single composite sync output to two separate H/V inputs is fine and often does what you want, in fact).
Of course, the above paragraph assumes your monitor wants "TTL Level" (5Vpp, 0V referenced) sync signals. If it wants sync signals at the same amplitude as your video signal, you have to get slightly more creative (74LVX86 will often work, depending on the amplitude of your video signals), especially since PCs always output TTL level sync.