Multiple content types from Views module, not old nodes from one of them
To filter out content that has a "final date" that is in the past, you can set up a filter as follows on the view for this content type. 

Filter on "final date" "is greater than or equal to" an offset date of "+0 minutes".

Screenshot below shows the final setting (in the screenshot example I used the "Post date" as the date field to filter on, but in your case it should be "Final date.")



Alternately, you could look at the Scheduler module and instead of using a "Final date" field, schedule these nodes to unpublish at a certain date. Depends on your use case.