Yes, you will reboot the Mac while pressing the Option key. Then you will select the usb drive as the startup disk.
If I recall correctly the installer will launch and run... if not, it'll be clear that there is an installer that you'll want to launch. It's been some time since I used a bootable installer myself.
EDIT - Actually, after you've booted the mini with the installer, you'll need to launch Disk Utility and proceed to erase the internal drive before you reinstall the OS. In ** you'll need to select the internal drive device, not just a volume, and erase that. In **, click View > Show all Devices. Then select the top most drive device, usually called "Apple Media..." or something similar. Use the APFS format, the GUID partition scheme and name the drive "Macintosh HD". When ** has finished that, you should also run the First Aid routine on the internal drive. That's also a function of Disk Utility.
When the First Aid scan reports all okay, then you can quit ** and launch the Reinstall macOS Mojave application from the macOS Utilities screen of the bootable installer and proceed with reinstalling the OS.