MacOS – How to know what to restore from system files

You should be able to connect it to an external monitor. All you need to do (in principle) is this:

1. Close lid, let computer go to sleep
2. Connect both the power supply and external monitor

About a second after you connect the external monitor, it should wake up and direct all output to the external monitor. (It will only do this if the lid is closed, external power is connected, and something is connected to the video output port.)

Note that depending on the speed of your computer and other factors, it may take a little while for anything to show up on the screen. Give it at least a minute if you don't see anything at first, before assuming something didn't work.

Apple provides no interface to pull individual items from the backup. Whether through iTunes or iCloud, backups are intended to be restored as a whole, replacing the content on the device at the time of the restore.

Is it possible that the image is also in your Photo Stream? That's probably the best possible solution if you do not wish to go through the steps to restore the backup to a device (your device or another one) to retrieve the photos you are missing (and then restore things again from a more current backup if desired).

If you do need to restore the whole backup, erase the phone (Settings -> General -> Reset -> Erase All Content and Settings) then you can restore from iCloud. When restoring from iCloud, you can choose from multiple recent backups.