I can attest that this AUR package, which is the one indicated in the wiki, works fine on EndeavourOS, but I had to add the gnome key wallet (IIRC) to handle the VPN keys. I think that was fixed.
Reading rorschac's comment I assume both OpenSSL and wireguard are already installed on CachyOS, or anyway pulled by the aur package.
If you want to make sure you can install them explicitly before protonvpn:
paru openvpn wireguard-tools
or using yay or the vanillapacman -Syu --needed openvpn wireguard-tools (it will sync and update the system too) or how it is suggested for CachyOS to install packages. I repeat I've no direct experience with that one.
If you are scared to mess things up you can always spin up a VM with CachyOS and try to install it inside that. If it all works you can then do the same on your main OS.
As a general advice, only run in your shell commands that you are sure about.
I spent hours trying to set up the GUI and couldn't get it to work. Tried the AUR, Flatpack etc and no luck. In the end I just set up the command line version of open VPN with an Alias.