While it makes sense now, it is not so easy to achieve, however the following may work. It extracts the attachment to a temporary message file on the local hard drive, then sends that file to the recipient of the original message (presumably you) whereupon it should arrive back in your inbox. Unfortunately I cannot find a way to skip the sending.

Select a message with the attachment and run the test macro and let's see what happens. If it works you can run the main macro from a script associated with a rule that identifies the original message.

Option Explicit

Sub TestMacro()
Dim olMsg As MailItem
    'On Error Resume Next
    Set olMsg = ActiveExplorer.Selection.Item(1)
    SaveAttachment olMsg
lbl_Exit:
    Exit Sub
End Sub

Sub SaveAttachment(Item As Outlook.MailItem)
Dim olAtt As Attachment
Dim olMail As MailItem
Dim FSO As Object
Dim tmpPath As String, strName As String
    Set FSO = CreateObject("Scripting.FileSystemObject")
    tmpPath = FSO.GetSpecialFolder(2)
    If Item.Attachments.Count > 0 Then
        For Each olAtt In Item.Attachments
            If olAtt.Type = 5 Then
                strName = olAtt.fileName
                MsgBox tmpPath & "\" & strName
                olAtt.SaveAsFile tmpPath & "\" & strName
                Set olMail = Session.OpenSharedItem(tmpPath & "\" & strName)
                With olMail
                    .Display
                    .To = Item.To
                    .Send
                End With
                Kill tmpPath & "\" & strName
                Exit For
            End If
        Next olAtt
    End If
lbl_Exit:
    Set olAtt = Nothing
    Set olMail = Nothing
    Set FSO = Nothing
    Exit Sub
End Sub