Exchange 2010 on 365: Converting a Remote Mailbox to Shared

Open Active Directory Module for Windows PowerShell

Run the following commands to connect with Exchange Server

$LiveCred = Get-Credential

$Session = New-PSSession -ConfigurationName Microsoft.Exchange -ConnectionUri https://ps.outlook.com/powershell/ -Credential $LiveCred -Authentication Basic -AllowRedirection
             
Import-PSSession $Session

Then run the following, substituting the user ID with that of your Remote Mailbox

Set-Mailbox -Identity "USER@yourcompany.com" -Type shared