I am going to a different country for a few months and I have a system there with a i5 2400.
My question is how big would the bottleneck to the GTX 1080 would be.
It will but depends on the game. I have a 2500k and it must be oced for most games because of frame dips. If those dips are fine then you could say no.
Also that's the opposite of what happens with higher res. Higher res shifts the bottleneck to the gpu not the cpu.