backing up iphone with no screen

No, but if you had it set to backup to iCloud then it may have done that automatically.

How to back up your iPhone, iPad, and iPod touch

https://support.apple.com/HT203977

"Automatically back up with iCloud Backup

To let iCloud automatically back up your device each day, here's what you need to do:

Make sure that iCloud Backup is turned on in Settings > [your name] > iCloud > iCloud Backup.

Connect your device to a power source.

Connect your device to a Wi-Fi network.

Make sure that your device's screen is locked.

Check that you have enough available space in iCloud for the backup."

If you were using iCloud to sync photos and videos it would be in your Photos app in iCloud which you can view on a computer browser logged into iCloud.com.

you could back your phone up using a computer.

That might work as long as it doesn't ask him to type a passcode or agree to trust this computer, or if he can even see it saying that.

Thats very true.