The new GPU will runs at x8 speed. However, there is a problem like that the new GPU may not work at full performance. CPU power may not be enough for the GPU. It may be better to choose a lower GPU to avoid bottleneck. Because CPUs with only x8 speed support are very weak.
Im planning on upgrading to a rx 580 , but yor point is that i should aim for a low performance card since most of the performance is going to waste? Also i wouldnt consider a ryzen 2400g a weak cpu or some sort , infact the excact opposite. If what you are saying is true i will have to switch out a cpu too, which will be over my budget.
It seems you would get double the performance going from 8x to 16x, in reality that isn't really the case. It is more about saturation of the bandwidth being used. In all probability you will see more like a 0% - 10% difference in speed. Yes your new RX 580 card is absolutely worth putting in that system and will provide better performance. Assuming that the CPU is not a bottleneck to begin with? As you only questioned PCIe type and not what CPU you have I can't answer that. Honestly though as long as you have on a fairly current system your CPU is very unlikely going to bottleneck an RX 580. That changes quickly however if you go to a flagship card, say like RTX 2080Ti. Again though you don't have that and it should not be an issue.
This site has a bottleneck calculator you can input your cpu and gpu to see if it will be an issue:
Another tip on that new RX 580, go into Wattman and set your power limit to 50. It greatly helps the stability of the card.