Is there any other tool or work-around in Outlook to backup emails if the exchange server is configured to prohibit archive emails in .pst? I am not sure what would happen if ignoring the rule and keep archiving emails to .pst. Maybe it won't work at all, or exchange server might still can delete the old emails even it's already archived. So I would like to find a more reliable solution.
Not sure how the serve is limiting PSTs. An administrator can prohibit use of PST files in Outlook or disable auto-archive to PST or disable use of an Archive mailbox. In any case, you may still be able to export to some extent.
From Outlook click File -> Open & Export -> Import/Export to start the export wizard. You can try to export to PST. Worst case would be exporting to CSV. It's not going to maintain full integrity (formatting, for example) when you import from CSV later. If there were just a few item you wanted to save you could just drag them to the desktop and save as individual items.