You'll get an excellent boost in performance, even with the i3.
From what I've seen, read, heard, and experienced, the idea in 'bottlenecking' performance has been greatly exaggerated by some.
Oh, it definitely does exist, but I've seen people extol a 3-5fps difference as 'definitive proof' that they had a bottleneck with their previous CPU when that 3-5fps is well within the expected margin of change for an OS that's doing dozens of other things in addition to running a game.
Now, with that said, the potential boost in performance from removing a bottleneck increases with the difference in performance between the two CPUs (or GPUs) in question. Your CPU is definitely an older light-weight so I would suspect that it is bottlenecking a bit in CPU intensive games.
In short, get the RX 580 and enjoy a great boost now. Get a CPU later and enjoy another boost in some of the more CPU intensive games.