Jumping from a i3-4160 to i5-4590 will definitely give better performance as for starters you are gonna get 2 extra cores with 6mb cache instead of 3. So u will definitely see better FPS in games like CSGO(which utilize CPU more)
He'd also have to buy a new motherboard and more importantly, RAM. Which is still expensive as shit. Or he can spend a tiny amount of money on a chip he just can directly swap without spending 8x the amount on what you're proposing. If he was gonna spend over 350euro he'd probably just save up and buy a new system altogether.
Hmm you'll get a boost in some CPU demanding games but don't expect much on most games. I used to have that exact same GPU and I did the exact same thing with upgrading my CPU but other than CS:GO I did not notice such a huge improvement at all.