As far as I can see on the spec sheet for the PC, it is equipped with two sticks.
I've been playing PUBG for a long time with 8 GB RAM, on i5-4670k, 8GB, 1060-6GB, and after having upgraded to 16GB, I see no higher avg. FPS. But the main advantage of more RAM, for me, has been the reduction of occasional stutters, when loading new content, e.g. when entering a room. In my experience, PUBG is very playable with 8GB.
I'm not that much into the AMD platform, but if RAM speed has a big impact on performance, you could consider getting faster RAM or just see if you can OC the RAM you've got already.
I see no reason to upgrade your RX580, as it would take quite an investment to achieve a substantial lift of performance. It is already a very capable card.
As far as I can see on various tests, an upgrade to a faster CPU seem like the most likely place to gain performance. To get a hint about the load on the CPU, you can launch task manager and watch the load on the cpu. Right click on the graph and choose "Logical processes" to see if one or more cores are maxed out. If this is the case, it is almost guaranteed that a faster CPU will help.
The R5 2600 or 2600X should provide you with enough oompf to deliver a consistent high fps output to your GPU. (Higher GHz, more cores and larger cache than your current CPU. All are things that matter)
If, of course the new CPU isn't being held back by slow RAM. But hten again, I can't really brag about being that much into the Ryzen/RAM speed performance.
And of course you have to be aware of the usual spuspects, such as not gaming over WiFi and not having stuff running in the background.
Also be aware that instead of just choosing a general "Medium" setting in the game, play around with the individual settings.
Note:
The above is about getting max FPS from your system, not necessarily the most beautiful graphics.
Edit:
All the above relates to playing at 1080p. If you play at 1440x2560 or 4k then a faster GPU will definitely help.