Smells funny to me. If he has done this before I would think he would know the Fedex hours. Seems like he would not be showing up to Fedex at closing to send a live animal overnight, Im sure that if it was me I would send them bright and early so they would for sure make it overnight.
Actually most people do drop them off right before last pick up. That way the animals are in the box for the least amount of time possible. I'd rather miss a ship date than have a DOA animal every day of the week.