Well, I can't give you the exact lowest/cheapest possible configuration which is what I guess you're looking for, because it's more about how slow/fast it will work, rather than if it will work. It will work fine on most computers from 2011-2012 or so I believe, but it might be on the slow side. Maybe unbearable.
I can tell you that the slowest computer I have used for 4k is a 2013 MacBook Pro (16 GB RAM, 2.3 GHz i7) and it has worked really well. But it can always go faster. With After Effects, the CPU and RAM is what you should be concerned about, not so much the GPU.
MacBooks and MacBook Airs (even brand new ones) will be considerably slower than old MacBook Pros.
Furthermore, Final Cut Pro X is MANY TIMES faster than Premiere and After Effects, but of course a bit limited in what you can do.
For serious single clip editing, After Effects is, in my opinion, the way to go.
On the 2013 MacBook Pro, editing several 4k clips into a film with basic color correction in FCP X works in real time, while a 10-second export in After Effects of a single clip with color correction, grading, and denoising may take 30 minutes to export.
It all depends on what you need to do to the clips.