what do you mean? to upgrade GPU or CPU? i was thinking to buy a new CPU like a i5... not very expenssive
About the CPU what i did was explaining how the bottleneck can scale with resolution in favor of the CPU rather then the GPU, lets see how can we make this more simple:
Let's say person A has a decent 1080p gaming PC. Now Person A will use max settings for all of his tests, the only thing he will change is the resolution. At 1080p in games let's say he has constant 100FPS and his GPU and CPU are in sync neither bottlenecking each other. But if Person A lowers the resolution from 1080p to 720p and plays again his even though his GPU is capable of doing 180 FPS it can only do 140 FPS because the CPU can't keep up calculating at that much speed for more then 140 FPS, thus a CPU bottleneck is created and the GPU has some breathing room. But if Person A does in reverse, going from 1080p to 4k, this time the GPU will be fully loaded doing just 45 FPS, and even though the CPU could do more calculations (up to 130 as we seen at lower resolutions) it doesn't have a way to do them thus now the CPU is having breathing room.
To go further in depth since games are complex programs, not simple constants, the loads on the GPU and CPU will vary depending on things rendered on screens, for example: when there's a lot of NPC's with their instructions for various actions it will be more strained for the CPU, meanwhile if the scene relies more on foliage such as detailed trees or grass things can get more heavy for the GPU, of course the opposite applies, if there's places with few NPC's the load on the CPU will be lessen, and if there's a place with almost no foliage the strain on the GPU will lessen. Because of things like this frames can get chaotic, jumping up to 90 FPS for example in a desolated place and dropping to 45 in crowded areas. Such situations can cause stutters.
Basically, very roughly this is the theory behind bottlenecking. So in other words, getting back to you, just in case you had a higher then 1080p screen i said that you can lessen the bottleneck on the CPU by increasing the resolution. Or you can use one of the V-Sync options, thus the frames will be limited to your monitor's refresh rate and the jump between minimum and maximum frames wont be that severe. Again though your choice was the right one, the RX580 in other words a graphics card was the best upgrade path now, it can benefit your current PC even if the CPU will slow it down a little, and even if you want better performance from other parts the graphic card can just be used together with a different upgrade (ie even if you would decide to upgrade the whole machine).
I hope i made things more simple now.
PS: Yes, an I5 would do a better job keeping up with the 580, also there are games which are designed for 4 threads now days, your cpu stuttering just because its limited to 2 threads, like Skyrim, some of the Assassins Creed games, Fallout 4, Dragon Age Inquisition, etc.