How do I recharge my mouse using mr computer

If you shut down the Mac the ports will no longer be active therefore you will have to plug in your mouse to the Mac and let the Mac sleep overnight. That will be more than enough to recharge the mouse.

Or get a charger module like those that came with earlier models of iPhones and then you can rechard the mouse with the Mac off..

Welcome to the Apple Support Communities.

Please see: Recharge the built-in battery in your Apple keyboard, mouse, or trackpad - Apple Support.

Cheers,

Jack

On an M1 iMac a charging cord comes in the box and this fits in the Thunderbolt socket. On other iMacs you need to source the appropriate cord.