How do I lock screen to back up? Then how do I unlock it?

It doesn't need to be locked if you are doing the "Backup Now" at Settings > [username] > iCloud > iCloud Backup.

For automatic iCloud backups you can either lock it by using the button on the right side of the phone or let it lock automatically via the time chosen at Settings > Display & Brightness > Auto-Lock.

Time to backup is dependent on the amount of data and the speed of your wi-fi connection. Mine will backup in minutes but I don't have much to backup (e.g. my photos are included in iCloud Photos and therefore are not part of the backup).

Press the side button to lock the screen. Or wait for the lock screen timeout. The phone must be connected to power and Wi-Fi for automatic backups, and it must be turned on. You can also go to Settings/<your name>/iCloud, tap on Backup, then Back up now.

You unlock it with Touch ID or by entering your passcode.

It sounds as though you've never setup 'Face ID & Passcode', which is fine, because if you're backing up to iTunes on your computer (Mac or PC), you'd have to unlock the phone to do the backup. Only when you do an automatic iCloud Backup do you need to lock the screen.

See the articles below:

https://support.apple.com/en-us/HT204060

https://support.apple.com/en-us/HT203977