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:
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.
Get-Mailbox -ResultSize Unlimited | Get-MailboxPermission -User "AD_user" | FT Identity, AccessRights, Deny