How to run script as another user without password?

try running:

su -c "Your command right here" -s /bin/sh username

This will run the command as username given that you have permissions to sudo as that user.

Call visudo and add this:

user1 ALL=(user2) NOPASSWD: /home/user2/bin/test.sh

The command paths must be absolute! Then call sudo -u user2 /home/user2/bin/test.sh from a user1 shell. Done.