You will gain more performance if buy new gpu instead of cpu. My friend just bought new pc with Radeon RX 580 and Ryzen 5 2600 and he can run the game on high settings above 60 fps easily. I think that if you would go just for the gpu you should stay above 60 on lowest settings. Upgrading you cpu will require from you also to buy new motherboard and ram memory. Not sure about the prices in country you are but Ryzen 5 1600x + RX 580 also will allow you to play the game even on high settings. You can check how much it will cost you (if you can afford RX 590 that would be even better for you)

EDIT: Just checked on Amazon, RX 580 is for 219$, RX 590 270$, Ryzen 5 2600 is even cheaper than 1600X thats even better for you if you decide also to go for cpu (149$). Motherboard for that cpu - Gigabyte B450 AORUS ELITE 100$ and 16GB DDR4 3000MHz - G.SKILL 16GB 3000MHz Aegis CL16 ~100$. You can start with gpu and start saving for other parts

EDIT 2: Clean your actual pc if you havent for a long time, high temperatures can cause your gpu and cpu work on lower clocks