TLDR:
Bottlenecking of the RX 580 is inevitable if wanting to play at higher resolutions, and higher graphic settings.
But is also dependent on the video game being played.
Suffice it to say, you will need a CPU upgrade at some point since that i3 is the bare minimum for modern gaming.
Doesn't mean you go and opt for the older and inferior graphics card.
Full Version:
you'll be fine for the most part.
Understand that your CPU is on the more low spec side of things period.
And so you'll get whatever performance you'll get from it.
If it turns out that the i3 doesn't allow the 580 to perform at it's peak, then that's just how it'll be, but you'll at least be getting everything you possibly can out of the hardware.
And if it is a bottleneck, then all you'll have to do is upgrade the CPU later on and that'll fix that.
But understand also that a "bottleneck" (as the term is thrown around a lot like a buzz word) is simply implying that one said component is not allowing another said component to perform at it's absolute peak performance (i.e. what it's capable of at max)
So to clarify, IF your i3 holds back the RX 580 from being utilized by a video game to it's maximum capabilities and performance, that doesn't somehow mean getting an older and inferior graphics card would somehow give you better performance.
Older less powerful GPU is still an older less powerful GPU.
The only "effect" you'll notice is frame rates, and stutter in games.
The only way you'll ever notice frame rates is if you look up performance benchmark videos of your exact model of RX 580 and it ends up turning out that you don't get as many maximum frames per second.
The only time you'll get stutter is from the CPU if the CPU has issues or something doing it's job correctly.
Suffice it to say, just get the newer and more powerful GPU.
And if it turns out that your i3 is keeping your GPU from performing to it's fullest potential by more than 5%, then consider upgrading the CPU later on when you can to end up getting the most out of it that you possibly can.
Also keep in mind that it can both matter and not matter if you have a low spec CPU or not because not all video games are created equal, and not all video games require as much CPU horsepower as others to run properly.
Lastly, as a general rule with "bottlenecking" it's always better to have a GPU bottleneck.
To clarify that, what that means is that your CPU has more to give than is being used, and the reason more is not being used is because the GPU is giving all it can creating this "bottleneck" as it's known.
This is a good thing, because it means that you are getting the most out of the graphics and CPU hardware that you paid money for.
So for future reference, when it comes to building a PC, if you already know what kind of GPU will be going into your system based on budget, try to get a CPU that doesn't equally match it, but has more headroom, so that you not only get the most out of the GPU when you build said PC, but then the CPU will not be a bottleneck when upgrading the GPU later on in the build's life span.
That was a long winded way of saying:
It's definitely possible the i3 you have will bottleneck the GPU from being utilized to it's fullest, but only in games that require more from the CPU than the CPU can give in order to allow the GPU to be used to it's fullest. That, and also the i3 you have is an i3 which is pretty much as low end as it gets while still being suitable for a more modern gaming PC. But at the same time helping to clarify the whole "bottleneck" thing and give you information that will help you in the future.
But yeah, low end. while it is hyperthreaded, it's still a dual core CPU.
Just two cores. that's it.
That's not much.