I have an RX480 8GB + i5-3570K @ 4.2GHz + 1440p 48-144Hz FreeSync monitor (Samsung C27HG70). What games are you playing?
It's difficult for me to quantify how
much my CPU is limiting my frame rates, but I will say that an RX580 is probably not what most people would consider a long term solution for 1440p gaming in the most modern demanding games. That being said, I'm able to stay above 70fps on most middling games and over 100fps on lightweight games (Doom included) all on High to Med-High settings. The problem with reviews is that the majority of them test games with video settings on Ultra. While it does establish a "worst case scenario", it's not practical since Ultra settings are known to offer nearly unnoticeable improvements in visual quality while taking a hard hit to performance [compared to "High"].
Monitors generally last through multiple GPUs. If there's a model out there you're itching for, go ahead and get it. Just pay attention to the lower limit refresh rate and get something with LFC. I've been running my C27HG70 for a little over a year and although I'll admit my RX480 isn't taking full advantage of the game settings or frame rates I could get with a more expensive GPU, I'm still able to enjoy all the games I play @ 1440p.
[Add] You could also try
AMD Virtual Super Resolution on your current monitor to simulate a 1440p load.
Here's some instructions