Yes, its effectively the same as wiring 4 buttons. On that wiring harness, one wire is the common ground, the other 4 are up, down, left and right. Wire it up to the arduino the same way as in the tutorial you linked.
I don’t see why you would want to use a joystick for midi as you are limited to what you can activate simultaneously using it, but its your custom midi controller so you do what you like with it. I made an arcade stick with cherry keyboard keys on it for someone this year so a midi joystick isn’t the weirdest custom thing people have wanted.
I recently created this tool. It maps XBox 360 type usb joystick or mouse/trackball to MIDI Sysex. I then us Bome MIDI translator Pro to remap it to the MIDI message of choice. I runs on Windows 10.