Twice a special game case has popped up but I have been unable to participate. How do I get in the..

Since you can't predict when the bonus cases pop up, it's mostly a matter of luck whether you'll be able to complete it. It might help to be prepared as much as possible to complete tasks in general, such as saving your Estate coins, saving your tips, not renovating landmark, not sending Jack on a delivery, not earning the seasonal set grand prize until the end of the month, etc.