Is there a way to log in on a Mac Mini without a Mouse?

Use the left or right arrow keys to get to the desired user, press Return to select it, enter the password, and press Return to log in.

Your keyboard can be used as a mouse with what Apple calls “Mouse Keys” mode:

Control the pointer using Mouse Keys on Mac - Apple Support

To toggle the mode: Press the Option key five times to toggle Mouse Keys

See Mac keyboard shortcuts - Apple Support

and figure it out…