My question is: can I wirelessly connect my Xbox controller to my PC and use it to play Fortnite wirelessly?
I don't know which controller you have. But I run my XBox One controller wirelessly without any problems.
I have a standart xbox one controller with bluetooth
If your PC has bluetooth: yes.
I use this adapter:
Yes that works according to my experience. Simply pair with Bluetooth in the settings. But you can also use a cable