A public events administrator maybe given a number of roles. Currently these roles are only assignable by a super user.
How to assign roles for "Public Events" and "Content Admin"
Switch to "Users" -> "Manage Amin roles", click on edit for the user, set the checkbox and update,
How to assign roles for "Approver"
This is calendar suite specific. Switch to a group that manages the calendar suite then select "Calendar Suite" -> "Mange preferences" and add the userid of the user to the list of approvers.