How to let Muon update manager run without entering a password?

try with configuring your crontab file, to make updates automatically, as root user, example:

in terminal write (as root loged user):

nano /etc/crontab

Then, for example, if you want updates every single day to perform at for example 14:30 o'clock at end of that file write:

30 14 * * * root apt-get update

and simply save changes.

That can help, and to don't worry about everyday updates, as it do that automatically at given time and given user and given command.