Q:

Is it possible to communicate an everyday computer with the ARINC bus?

Lately I've been wondering if it's possible to interface an everyday computer with the ARINC buses (either 429 or AFDX).
This would be very useful for simulation, testing and prototyping purposes. For example, we could write a computer program that sends ARINC labels to the bus and see how the avionic systems react to it.
But I think that the greatest challenge would be in the physical layer.
The image on the left shows some connectors that we can typically find in everyday computers. The image on the right shows the connectors that we can find in the front panel of an avionic system.
             
    
At first glance making such a connection does not look straightforward.
Nevertheless I've done some research and I have found devices which seem capable of interfacing a computer with the ARINC bus via the USB port:

These pocket-sized USB adapters are the easy and portable way to interface your computer to ARINC 429 and ARINC 717 Avionics
  databuses.
Source: Astronics

So connecting a computer and avionics is possible in theory. Still, I don't see how I would be able to connect this USB-ARINC adapter to the avionics box, since the connectors are different.
So there must likely be an access point somewhere in the avionics where this adapter can be connected.
This is all I could find.
So again,

Is it possible to communicate an everyday computer with the ARINC bus?
If so, what is the access point where we would connect the wire?
Would there be any other challenges I didn't take into account?

A:

Is it possible to communicate an everyday computer with the ARINC bus?

Sure. there are appropriate controllers to be plugged in a slot - or even as USB, like the one you found. After all, basic USB is alread way faster than ARINC 429. 

If so, what is the access point where we would connect the wire?

Whatever connector the interface card/box offers. More often than not this may require some special-to-type cable. If you're able to handle a soldering iron and/or crimp pliers, building one is a reasonable job. If not, that's where you may ask the interface supplier to take even more money to sell you a fitting one. 
On ARINC 664 (ADFX) it's even less an issue, as there are as well equipment using standard plugs.

Would there be any other challenges I didn't take into account?

Software. As always it's the software there the main cost and effort is hidden. While basic drivers are usually included (for ARINC 429) or default (ARINC 664), any layer above will be specific to the task you're intended to handle - and usually not exactly low priced.
For ARINC 664 it depends even more on the task intended. While basic communication runs on standard Ethernet layer, 664 adds a quality of service at layer 2. Something you OS default driver may not support. Ofc, one may (buy and) install a driver that does, still this might bring trouble on the 'regular' networkside. Due this (and to avoide collisions) adding a separate interface card with fiting drivers may be advised. 
ARINC 664 opens another can of worms depending on the application intended. Basically it is a virtual implementation of a bunch of ARINC 429 busses over Ethernet. To make applications work some translation layer may be needed.
Bottom line, buying and pluging an interface in/to your PC is the least complicated step here. And everything beyond that needs in depth knowledge of what is intended to be archieved.

