When does the 'aqiqah for a child born after the Asr prayer on Friday, September 7, 2007, end, and is it still permissible to slaughter it, taking financial circumstances into consideration?

The Sunnah is for the 'Aqiqah to be on the seventh day. If it is missed, then on the fourteenth. If that is missed, then on the twenty-first. Weeks beyond that are not considered; rather, it is recommended at any time one is able to perform it. Therefore, your brother should perform the 'Aqiqah for his child whenever he is able to do so.