Run a shell script as another user that has no password

You can do that with su or sudo, no need for both.

sudo -H -u otheruser bash -c 'echo "I am $USER, with uid $UID"' 

The relevant parts of man sudo:

-H   The -H (HOME) option requests that the security policy set
     the HOME environment variable to the home directory of the
     target user (root by default) as specified by the password
     database.  Depending on the policy, this may be the default
     behavior.

-u user     The -u (user) option causes sudo to run the specified
      command as a user other than root.  To specify a uid
      instead of a user name, use #uid.  When running commands as
      a uid, many shells require that the '#' be escaped with a
      backslash ('\').  Security policies may restrict uids to
      those listed in the password database.  The sudoers policy
      allows uids that are not in the password database as long
      as the targetpw option is not set.  Other security policies
      may not support this.

su can only switch user without providing a password if you are root. See Caleb's answer

You can modify the /etc/pam.d/su file to allow su without password. See this answer.

If you modified your auth file to the following, any user that was part of group somegroup could su to otheruser without a password.

auth       sufficient pam_rootok.so
auth       [success=ignore default=1] pam_succeed_if.so user = otheruser
auth       sufficient   pam_succeed_if.so use_uid user ingroup somegroup

Then test from terminal

rubo77@local$ su otheruser -c 'echo "hello from $USER"'
hello from otheruser

If you want to use su instead of sudo, I believe you can use something like this:

su - <username> -c "<commands>"

• - will simulate a login of the specified user
• -c tells it that you want to run a command

ps. Unfortunately I'm not able to install ruby using rvm with this method, but that's probably not related.

The answers above are really useful to me but to answer the actual question...

How can I affirm that the script is really running under that user now?-

Use:

ps -ef | grep <command-name>

The output should include your script and the actual user executing it. People on BSD-like systems, e.g. MAC can find similar information with:

ps aux | grep <command-name>