sudo access for a user to read mail files
 

What do I do to give a user the ability to read mail files in the /var/spool/mqueue directory

In the first place you won't use sudo for that, this is the wrong tool for the job.
This is a job for file permissions. On my Slackware systems the directory /var/spool/mqueue is owned by the user root and the group mail. So the easiest solution would be to add the user to the mail group to give him access to that directory.