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No, this cannot possibly be implemented. Moreover, if you are our client, you should NOT be using Joomla's automatic updates. Let's elaborate.

How the different update methods work

The backup cannot run "magically". It needs to be told to run, and it takes several page loads to complete. The Backup On Update plugin essentially intercepts the button in Joomla Update which tells the Joomla Update component to download the backup. Instead, it sends you to Akeeba Backup which takes a backup, and then takes you back to the URL which tells Joomla to download the update. That page sends you to the Joomla Update page which extracts the update. When that's done, it takes you to a page which runs the post-update code.

When you are using automatic updates none of that happens. It is NOT even Joomla installed on your site updating itself. What actually happens is that you register your site to a service operated by OpenSourceMatters Inc (the non-profit running the Joomla project), hosted on the joomla.org infrastructure. When there is an update available the service on joomla.org will instruct your site to download and install the update. There is no provision in joomla.org's service to run anything else before or after the update. Therefore, a backup cannot run as we are not given a "hook" to put a backup step in this process.

When you are using a third party service (such as mySites, or Watchful), or you are using an external self-hosted service such as Akeeba Panopticon there is explicit provision to run a backup before installing the update. What happens in this case is the following. The service becomes aware that an update is available for your site and it needs to install it. First, it runs a backup of your site using Akeeba Backup Professional. When the backup is complete, it asks Joomla Update to download the update package. Then, it uses either its own code or Joomla Update's code (depending on the service) to extract the update package, and run the post-upgrade code. Just like the Backup On Update plugin, there is an explicit backup step.

Which alternatives do you have

There are three alternatives for you.

One, is taking daily backups and HOPE that Joomla's automatic updates are not installed while a backup is in progress. You see, unlike any site management service out there, Joomla's automatic updates do NOT give you any control over when the update will be installed. It might happen at a very inopportune time for your site, breaking everything.

Two, use a third party service to install updates whenever it makes sense for your site, taking a backup before doing so. Remember to turn off automatic updates. This is the best option as you have control over when to install the update, and you have a reliable backup in case something breaks. As I said, you can always use Akeeba Panopticon to run your own site monitoring service without having to pay someone else to do that for you.

Three, go fully manual. Turn off automatic updates. When an update is available and you know your site is at a state it makes sense to take a backup and run the update, log into your site's backend and do the update. This solution makes sense if you only have one or two sites. Anything more than that and it makes a lot more sense using a service.

Why you should NOT be using Joomla's automatic updates

Joomla's automatic updates are NOT meant for the kind of site our clients have. They are meant for people who have created "set up and forget" sites such as very small business sites, simple personal blogs, social club sites etc. These are sites with minimal traffic / site visitors, and typically a part-time admin who can't be bothered to install updates regularly. The problem with those sites is that they tend to sit on vastly outdated versions of Joomla with known security issues for months at a time. This makes them ripe targets for attackers. They get hacked, and their owners blame Joomla instead of their own lazy management practices for their predicament. THIS is the kind of site Joomla targets with automatic updates. Any site which has more than minimal traffic, and an administrator actually paying attention –which of course includes the sites of our clients– should NOT be using this feature. Having updates installed haphazardly at random times of the day is very likely to break your site, and you may end up in a situation where you don't even have a suitable backup, losing you a lot of work and/or causing financial harm to the site's owner.