Run backup script at any time every weekend (using launchd?)

Launchd will run your jobs next time the mac wakes from sleep. So if you shut down your mac, this won't work. You will need to put your mac to sleep to get your daily script to run when the mac wakes if the last time it should have run was during the sleep duration.

man launchd.plist:

If the system is asleep, the job will be started the next time the computer wakes up. If multiple intervals transpire before the computer is woken, those events will be coalesced into one event upon wake from sleep.

Or in other words: You don't have to do anything specific, it's the default behavior anyway.

If you also want to run the command once after reboot, add

<Key>RunAtLoad</Key>
<true/>

to your launchd plist.

You can print the StartCalendarInterval dictionaries with something like this: for h in {0..4}; do for m in {0,15,30,45}; do echo "<dict><key>Hour</key><integer>$h</integer><key>Minute</key><integer>$m</integer><key>Second</key><integer>0</integer></dict>"; done; done.

You could also run the script every 15 minutes but exit it depending on the time:

h=$(date +%-H)
m=$(date +%-M)
[[ $h -gt 4 || $h -eq 4 && $m -ne 0 ]] && exit

Or run EDITOR=nano crontab -e and add a line like */15 0-4 * * * ~/bin/script.