Xbox – Far Cry 4 Downloading when I have the disc

this is kind of dependant on what the game does. some games have specific wallpapers for achievements while others use a simple image.

The custom backgrounds was part of the November 2014 update which was around the same time Far Cry 4 launched, so it's understandable they did not have specific backgrounds. It's also a third party game, so I'm not sure that they wanted to put in the extra work in creating images for each achievement.

You can set a custom background to whatever you want, it doesn't have to be an achievement. You can read up on a guide and get a template for custom backgrounds here.

but... to answer your question: no, there is no way to use 360 style achievements on the XB1

From that download link you have in your question, on the top right of the page you will see "My Xbox." Click it, and select "My Games." This will show you all the games you own for both Xbox One, and Xbox 360. For you, all your Xbox One games should only be the Games with Gold games since that's all you've ever purchased.