Stop the world I want to get off, eh?
I don't have any very good suggestions except to point out the potentially obvious note that all the messages aren't necessarily urgent. If animals want mates or a baby is born or someone dies from old age, there isn't really anything you HAVE to do about it. If you do what alloy suggests and lock the zookeepers inside, you can probably rest assured the whole zoo isn't running wild and just ignore the messages for a while. (EDIT: OK I guess I read alloy's answer too fast...he said you could ignore the messages too.
)
I had always thought, though, it would be kind of nice to have a staff member called a zoo manager that you could pay for just that purpose. He could be kind of a super-zoo keeper, and have an expensive salary, and have the power to do things like release animals into the wild, hire more zoo staff, and make other decisions, so if you did want to hire one of them and stop worrying about that stuff, you could.