At first update the old system.

pi@rpi0100:~# sudo apt update
pi@rpi0100:~# sudo apt-get upgrade
pi@rpi0100:~# sudo apt-get dist-upgrade
pi@rpi0100:~# sudo apt-get full-upgrade

Then replace buster with bullseye in /etc/apt/sources.list:

pi@rpi0100:~# sed -i 's/buster/bullseye/' /etc/apt/sources.list

And finally start the update and upgrade process:

pi@rpi0100:~# sudo apt-get update
pi@rpi0100:~# sudo apt-get upgrade
pi@rpi0100:~# sudo apt-get dist-upgrade

In case the last command causes the following error:

Die folgenden Pakete haben unerfüllte Abhängigkeiten:
libc6-dev : Beschädigt: libgcc-8-dev (< 8.4.0-2~) aber 8.3.0-6+rpi1 soll installiert werden
E: Fehler: Unterbrechungen durch pkgProblemResolver::Resolve hervorgerufen; dies könnte durch zurückgehaltene Pakete verursacht worden sein.

Try:

pi@rpi0100:~# sudo apt install gcc-8-base

And then continue with this:

pi@rpi0100:~# sudo apt-get dist-upgrade
pi@rpi0100:~# sudo apt-get full-upgrade

Time for a reboot?

pi@rpi0100:~# sudo reboot

Finished.