[Foivos]

How can you even install the update without access to the Internet?

For years, I can not do the automatic updates, because it always fails with an error message along the lines of "Failed to personalise software, check your internet!", even though I have a perfectly working Internet connection. The only way to update is with a live USB and an ethernet connection. Everything else fails.

[thisislife2]

You can manually download the updates from here - https://support.apple.com/en-us/docs for the older versions of mac (or check and download it manually through Software Update). Once the installation starts, the installer does some verification and / or data collection for which it requires the internet, and then starts the installation - at this point, just switch off the router.