100$ GPU would be plenty.
10600K(F), overclock it as high as you can, then cry because minimum fps are still going to be nowhere near 360 fps.
Not like it should matter with G-Sync anyway, unless you plan on paying for that and then not using it.
I assume you're talking about the PG259QN?
So you'd be spending half again as much as you planned for your build on the monitor?
And for what? It's an IPS so it's going to be the exact same thing as with the VG259QM/VG279QM.
Usable at lower than the advertised refresh rates, but if you actually max it out it starts smearing.
Those are officially 280 Hz.
240 Hz is fine:
Now 280 Hz not so much:
FYI what that means is the panel takes longer to change the colour than the frame is actually displayed. So it'll only reach that colour after the next refresh. If the colour stays the same. If the next frame has changed the colour again then it'll probably keep lagging behind.
Don't get me wrong, they're very good monitors at 240 Hz or 280 Hz with G-Sync/FreeSync enable since you wouldn't reach 280 fps most of the time and motion blur reduction with Sync enabled is nice, apart from some overdrive issues at very low refresh rates there's nothing wrong with them.
But I really doubt that the issues they have at 280 Hz are going to magically have been fixed at 360 Hz half a year later.
It'll be double the price for at best another 60 usable Hz, bringing it up to maybe 300 Hz usable, which you won't ever reach as a minimum refresh rate in TF2.
Ask someone with a 10600K or similar or 9600K/8600K or at least something in that ballpark to record their frametimes and check how much time is actually spent above 240/300/360 fps. And then ask yourself how much of that is staring at a wall. Unless you're only playing MGE any fight where you actually want/need the high refresh rate you're not going to get 360+ fps. So what would be the point?