It is very hard to schedule stuff to run on log-off/shutdown on Windows for two reasons:
- There are no events accessible to the user that you hook up to. This is probably because of the next one
- It is a shutdown/log-off and Windows is broadcasting messages left, right and centre announcing that so software can exit gracefully - not exactly the best time to start new tasks or process.
The developers here will probably be able to confirm, but I am pretty sure you may be able to tap into the Windows API and hook-up to something, however, that is likely not what you are looking for.
If I were in your position, I would create an orchestration script that takes care of stuff (i.e. close Outlook or log your mom off) and then runs Duplicati. You can also leverage the run-script-before and run-script-after advanced options and leave everything to Duplicati.
A combined approach may be best - create an orchestration script that will log-off your mom and run Duplicati and use run-script-after to shutdown the computer. You can then create a shortcut to the script and tell your mom that this is the new shutdown button.
Having said that, are you really sure you want her to keep stuff in a PST file? If her account is on Exchange or Outlook.com it may be better to just leave everything on the server or in the cloud. If her account is with another provider, the same is true but only for mail items, so I understand that you may need the PST for contacts, calendar, etc. Still, tough, there may be a better option…