Would this cooler work?

It will work but I highly advise against using the Corsair H80i. It performs about the same as a $30 air cooler so there really is no value in it. I'd suggest getting the Noctua NH-D14 instead.
 


Right....and it would take a bit of effort to install.....does that one install easier?
Btw I can't access the back of the mobo.
 


both coolers would require installing a backplate on the motherboard so either one you go with, you'll need to pretty much disassemble your computer and pull out the motherboard.
 
Thanks then I can't do that. Is it true that I don't have to get a new mobo until it breaks or my new cpu won't work?

I also broke one of tha golden pins (USB probably) like a year ago but it still works great and boots super fast on hdd. Would that matter?
 


No this is not necessary. Your case has an open motherboard hole so you only need to open the case side plate to get to the cpu backplate.