Note: This will be written as a stamina-kill guide. If you're not interested in getting his camouflage, feel free to use any weapon against him.
The Fury is a pain until you get his pattern down... and even when you do. I find none of the special camos up to this point will actually help.
Your main weapons in this fight will be thermal goggles and, if you managed to stamina-kill The End, the Mosin Nagant. The Mk22 will suffice if you don't have it.
When the fight starts, immediately run to your right, then forward to the end of the hallways. Once you get to the end, turn around, equip the thermal goggles just long enough to see if The Fury is in a particular hallway. Unlike most enemies, which show up white, The Fury shows up blue. With the Mosin Nagant, aim for the head and get a single shot off, then get as far out of the line as fire as possible before he comes blazing through. If you're using the Mk22, try to get three shots off so he falls over and gives you more time to run away.
Be patient with this one! Trying to get a shot off every time you see him is asking to be set on fire numerous times. Also, while the northern end of the halls are safe, don't stick around too long on the southern ends. There's no railing, so if he hits hard enough, he can send you right off the edge to your death.