Which part of a computer is critic to work on 4k footage
The hard drive is usually the biggest factor when working with big footage files.
Upgrade his workstation with a reasonably sized SSD, I recommend atleast 500GB to ensure you can work on multiple projects without moving files around.
The SSD speed of course is the important factor that would speed up the workflow, so be sure to not buy a cheap but big SSD but one that has fast read and write speeds. Read speed being  more important than write speed.

When working with Adobe Products, a mid-range GPU can help a lot aswell when doing a lot of color correction and other "effects" work. Many effects and plugins nowadays are GPU accelerated and will be even more in the future.
I recommend buying a mid-range consumer or "gamer" GPU instead of a so called workstation GPU. They are incredibly overpriced with nearly no additional stability and usually worse performance. You only benefit from them if you have very special needs (like 10bit displays or frame locking) and need certain warranties or support from the Manufacturer.

Also as AJ Henderson mentioned, you should have around 16GB RAM or 12GB on 3 channel CPUs. 8GB or less will perform horribly with modern Adobe Products, they use a lot more RAM to increase performance since they switched to 64bit.