Check Account Permissions to Mailboxes
If you want to check what permissions an Active Directory user account has on a specific mailbox, simply use the following cmdlet which will enumerate all the permissions the user has on that mailbox:
Get-Mailbox <Mailbox> | Get-MailboxPermission -User <AD User>
If, alternatively, you want to check which mailboxes that specific user has permissions to, you can update the cmdlet to:
Get-Mailbox -ResultSize Unlimited | Get-MailboxPermission -User <Active Directory User> | Format-Table Identity, AccessRights, Deny
However, please be careful as this cmdlet will enumerate all mailboxes in the organization. If there are dozens of thousands, you might want to target only specific mailboxes.