You can configure the synchronisation setting for the Outlook account in Thunderbird by right clicking the account and choosing to synchronise only the last x weeks of email.
Just to clarify how synchronisation works messages older than x days will still appear in the folder but they will not be able to be accessed while offline and require a connection to the server to download upon selection. This should not be an issue with what I assume is an always on broadband connection albeit with data cap.
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