The KB below will show you how to check to which mailboxes a specific user has permission to.
This can only be done using Exchange EMS.
Type the two cmdlet below and just change the Administrator user with the user you need to check.
$mailboxes = get-mailbox –resultsize unlimited
$mailbox | get-mailboxpermission –user administrator | where{$_.Deny –like “False”) | format-table identity, AccessRight, Deny