To grant full access to an Office 365 Mailbox you must first connect to the remote PowerShell, by typing the 3 commands in your local PowerShell:

$o365cred=get-credential
$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://ps.outlook.com/powershell/ – Credential $o365cred – Authentication Basic – AllowRedirection
Import-PSSession $session

After you are connected, you must run the following command to give Alan full access to Bob’s mailbox:

Add-MailboxPermission -identity Bob@domain.com – user Alan@domain.com – AccessRights FullAccess

If you wanted to give Alan full access to all mailboxes in your environment you would run:

Get-Mailbox | Add-mailboxpermission -user Alan@domain.com -AccessRights FullAccess