I'm running Octopi on a Pi 4B. A few days ago I read about a firmware update that gives little bit better performance in temperature terms.
Is it possible to make a firmware update without losing the functionality of the latest Octopi Image? And also what would be the safest way to execute this firmware update?
Well, your reboot should bring both services back into action, for what it's worth.
If you read this, you'll see that the people who make Raspbian have documented the procedure. To the best of my knowledge this will add the rpi-eeprom-update command line tool (having installed it in the first line). Additionally, it will add a service which runs it automatically but you can control that as described in the docs. Follow the links in that for more information.
If you do (accidentally) run rpi-update (bringing in the bleeding-edge next branch) it WILL automatically update the firmware plus overwrite the EEPROM part of that. If you need to come back from that (bricked Pi) then there's another procedure on there for creating a one-time-use microSD to boot the Pi and bring it the earlier firmware.
The question remains: is the EEPROM update service already in Raspbian Buster Lite from 2019? I had heard that it was. So I'm not sure as to the dating of the documentation. It's possible that this was from around Mar/Apr 2019 (in theory, before they included that). But I don't honestly know.
The only things you need to run for the USB/PCI-E firmware updates are:
sudo systemctl stop octoprint
sudo apt update
sudo apt upgrade -y
sudo reboot
Then sudo apt install rpi-eeprom
sudo rpi-eeprom-update -a
sudo reboot
this will update the eeprom (different from rpi-update. which as OutsourcedGuru said, shouldn't be used unless you are a developer or it addresses a specific issue you are having with the kernel.