I will be upgrading from 1.4.14 to 1.5.9.
PHP version: 5.6.40
Since this is such a big jump in versions I am not sure if problems can be expected.

Was planning on following these instructions:
https://www.zenphoto.org/news/installation-and-upgrading/#--upgrading

As mentioned on that page we don't test that specifially so we can't exactly tell. Probably the update itself will work.

The biggest problems will be custom/third party themes and/or plugins that may not be maintained and therefore compatible. Also things changed and some lugins have been removed. If it is an important live site, use a staging site first.

Also note that we are currently working on Version 1.6 we hope to release soon.

How about restoring if the upgrade is unsuccessful?

You could start moving from php 5.6 to php 7.4, then, if ok, from Z 1.4.14 to 1.5.9.

Moving from 1.5.9 to 1.6rc : as mentionned by acrylian, you will need to check your debug page to adjust / remove some functions in your theme templates.

Then, you will be able to move from php 7.4 to php 8.1, and will be ok !

> you could start moving from php 5.6 to php 7.4,

Correct, 1.4.14 should be PHP 7 compatible already. Not sure about 7.4. specifially as it was five years ago.

> How about restoring if the upgrade is unsuccessful?

As usuals create an backup of the database (use the ZP tool and a real db backup via phpmyadmin) and then re-install thte old version. In any case in case of an important live site create mirror install (= staging site) on the server first and try with that first. Fix all bugs there if possible and then upgrade the actual site.