Ahh I think I should have added in the Item.Save part:

Option Explicit

Private WithEvents inboxItems As Outlook.Items
Private Sub Application_Startup()
    Dim outlookApp As Outlook.Application: Set outlookApp = Outlook.Application
    Dim objectNS As Outlook.NameSpace: Set objectNS = outlookApp.GetNamespace("MAPI")
    Dim ShrdRecip As Outlook.Recipient: Set ShrdRecip = objectNS.CreateRecipient("g.clark@maritimetransport.com")
    Set inboxItems = objectNS.GetSharedDefaultFolder(ShrdRecip, olFolderInbox).Parent.Folders("Sent Items").Items
End Sub


Private Sub inboxItems_ItemAdd(ByVal Item As Object)
    Dim tmp As String
    With Item
        If InStr(UCase(.Body), "HI") > 0 Then
            .FlagRequest = "Follow up"
            .FlagDueBy = Format(DateAdd("d", 2, Date) + TimeValue("18:00:00"), "dd/mm/yyyy hh:mm")
            .Save
        End If
    End With
End Sub