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