Well the manual shows just 2 VGA ports without the "optional video unit".
It also has no speakers of any kind, but will drive 6/8 ohm speakers if connected to the unit.
The red(right audio) white(left audio) and Yellow(SD video) do not seem to have a place to connect to this TV.
You had mentioned that you actually had picture with this monitor. Does it have this "optional video unit" already installed?
The TV also has 2 3.5mm headphone jacks for audio input(one for each VGA port). You would need an adapter to get this to connect from the red/white RCA jacks. All this still depends on hacking passive speakers connected to the 2 speaker outputs on either side of the screen. The headphone jacks are inputs and can not be used for this.
In the case you want to try, first you should use a computer to confirm the monitor works including its audio input(you need active speakers 6-8 ohms and at least 10 watts).
This is the adapter they sell for such a use
http/www.xbox.com/en-CA/Xbox360/Accessories/CablesNetworking/Xbox360GVGSHDCable
Now that cable assumes normal RCA inputs for audio, so another adapter would also be needed. Something like this.
http/www.amazon.ca/Cables-Go-40645-To-Adapter/dp/B000J1H4VI
I am only listing those adapters because they are what my friends at Google come up with. I am sure you can get store brand parts that do the same thing locally if needed. The VGA adapter HAS to be made for xbox 360 as that is the AV port and not an HDMI port on the end.
An active HDMI -> VGA may work, but chances are it will cost more and may not also bring audio with it(depends on the adapter.).
If you do not want to deal with the TV speakers you can connect any old set of power computer speakers to the red/white cables and you should get sound.