No admin page on update Pihole v6.0.5 on RP Zero W

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What do you mean? Please elaborate.
The command to update Pi-hole is the same in v5 and v6: pihole -up

Each of these changes (and a lot more) were described in v6 release notes:

Pi-hole v6 now uses its own web server (embedded into FTL). Apache2 cannot execute the new web interface pages.

The installer identified there was something using port 80 (apache2) and used the alternative port 8080 for the new web server (pihole-FTL):

*** [ DIAGNOSING ]: Ports in use
...
[✓] tcp:0.0.0.0:8080 is in use by pihole-FTL
...
[✓] tcp:0.0.0.0:443 is in use by pihole-FTL
...
[✗] tcp:*:80 is in use by apache2

Also note that Pi-hole now offers https (port 443).

Currently you can access the web interface using one of the URLs below:

Note:

You are running an outdated and unsupported OS (Raspbian 10).