In Ubuntu, calendar displayed with date and time uses locale file to setup. If your locale has first day of week set to Sunday and you want to change it to Monday, it is possible without changing locale itself. I like to stay for various reasons on en_US locale, but I like my first day of week on Monday.

How to find out, which locale file is Ubuntu using right now, enter locale command?

$ locale

That will output:

LANG=en_US.UTF-8
LANGUAGE=
LC_CTYPE="en_US.UTF-8"
LC_NUMERIC="en_US.UTF-8"
LC_TIME="en_US.UTF-8"
LC_COLLATE="en_US.UTF-8"
LC_MONETARY="en_US.UTF-8"
LC_MESSAGES="en_US.UTF-8"
LC_PAPER="en_US.UTF-8"
LC_NAME="en_US.UTF-8"
LC_ADDRESS="en_US.UTF-8"
LC_TELEPHONE="en_US.UTF-8"
LC_MEASUREMENT="en_US.UTF-8"
LC_IDENTIFICATION="en_US.UTF-8"
LC_ALL=

That means, that my locale is en_US, settings for this locale are saved in file en_us in /usr/share/i18n/locales/ directory. 

$ sudo nano /usr/share/i18n/locales/en_US

Edit the en_US file and change first_weekday from 1 (Sunday) to 2 (Monday). Then generate locale files:

$ sudo locale-gen

Then restart unity-panel-service:

killall unity-panel-service