In the link, the seller is writing that the cabinet contains a MS9-29 15 kHz monitor. He talks about an arcade VGA3000 video adapter and also about a Radeon HD 4350 video adapter.
In your youtube video, the video signal seems to come from the pc motherboard itself. I doubt that can produce a 15 kHz (or 25kHz) signal that will keep that monitor happy.
besides that, plugging in a video adapter on that pc mobo without any mechanical support is a very bad practice. The video adapter needs to be aligned perfect veritically, or it will give you problems. So, you basically should have received one or 2 video adapters with the system, unless you just bought one "like" the link you posted, and not "the one" from that link. If you have the correct setup with one of 2 video adapters mentionned in the link, you will need to setup a system with a normal pc vga monitor first. Once you have that running, you can install the special arcade video drivers and you can switch the resolution to 15 khz. After the switch, the normal pc monitor likely won't give a picture anymore, but the arcade monitor in your astro should do so.
It's strange if you find a loose adjustment board in the cabinet, as it's normally part of the monitor chassis. Wthout it, you can't do anything with most monitor chassis. So, even if the cabinet got a different monitor, you still would keep the old adjustment board together with the original monitor from the cabinet. The seller showing pictures on a forum of a system setup on 15 or 25 khz, and saying that he doesn't have any technical knowledge to help, sounds a bit strange to me. I also don't understand why he wipes the harddisk clean before he sells the system, since that removes the option to test the system. (Unless that was the purpose) The video showing the floating components of the system speaks a bit for itself.