The RX580 can display up to 8K actually (even just max at 30FPS, but it can do 8K).
4K isn't the limit of the GPU, and the DP 1.4 can deliver 4K at 120FPS, which is way more than enough for daily use.
TBH, even a very old HD7000 series GPU can display 4K 60Hz without any trouble. So, for a much more powerful RX580, that's not stressful at all.
Of course, I am talking about normal 2D environment. For 3D gaming, 4K is very demanding, and RX580 is far from good for 4K gaming.
Before I upgrade to Radeon VII, I used the RX580 for a while, and it works very well in Mojave, very smooth. Even my display is a 3840x1080 32:9 monitor. I actually use 7680x2160 @144Hz virtual resolution to render a 3840x1080 HiDPI desktop (this provide better image capture, and much clearer text, etc). And a RX580 can do that without any difficulty.
So, if you only use it on normal 4K 16:9 screen, most likely you will only use up to 5K virtual resolution to render 1440P HiDPI desktop. That should be nothing for a RX580.