This is should be pretty much the same on Ubuntu 14.04 as it is on the latest releases - however, they maybe some differences.
The task is divided into two parts:
- To be able to sudowithout a password, add the user to thesudoersfile and allow them to not require a password.
- To run a job automatically, use cron.
sudoers
Add the user to the sudo group with
usermod -aG sudo <username>
Change <username> to the user that you want to run the job as.
To avoid having to enter the password, edit the /etc/sudoers file:
visudo
and add the following line at the end of the file:
<username>  ALL=(ALL) NOPASSWD:ALL
Again, change <username> to the user that you want to run the job as.
Note that a better approach would be to create a new sudoers file for your particular user instead, instead of editing the main sudoers file, like so:
echo "<username>  ALL=(ALL) NOPASSWD:ALL" | sudo tee /etc/sudoers.d/username
For more info see How to Add User to Sudoers in Ubuntu.
See also this question, How can I add a user as a new sudoer using the command line?
cron
cron is very powerful and there are a lot of different options. However, to accomplish your task, here are just the basic essentials:
To edit the crontab, use
crontab -e
or for a different user, use
crontab -u ostechnix -e
If you have never run crontab before, you may be asked to choose an editor the first time that you run it.
Then for a 3 pm job add the line
0 15 * * * <command-to-execute>
So in your case use
0 15 * * * sudo php artisan backup:run
Save and exit to editor. Then to check the crontab, use
crontab -l
If you want to change the time the fields are as follows, from the cron man page.
       The time and date fields are:
              field          allowed values
              -----          --------------
              minute         0-59
              hour           0-23
              day of month   1-31
              month          1-12 (or names, see below)
              day of week    0-7 (0 or 7 is Sunday, or use names)
       A field may contain an asterisk (*), which always stands for
       "first-last".
For more info about crontab, see A Beginners Guide To Cron Jobs, or type man cron.
See also this question, How do I set up a Cron job?