How to set up and register new Mac mini with my new monitor

What type of wireless mouse?

If its bluetooth, then you will need a wired one.

If its one with a usb receiver, just plug in the receiver into a USB port on the Mac.

Otherwise as mentioned, borrow a wired mouse for the setup.

Dig out, barrow or buy an inexpensive USB Keyboard and Mouse to setup the new Mac mini.

Then once the Mac mini is setup, connect your wireless Keyboard and Mouse.

Can you not click on the "Continue" arrow there?

I think you meant borrow not barrow https://www.merriam-webster.com/dictionary/barrow...