You are automatically a member after cruising once. However, they are kind of inconsistent as to how soon they send out your membership information. I've been told that people sometimes have to call in and get their number. Others (myself included) received their membership packet a month or so after cruising.
All you really need is the membership number. The Castaway Club info number is 1-800-449-3380. If you call them and let them know when you cruised, I think they'll be able to get your number for you (for use when booking a new cruise) and possibly to expedite the shipping of your membership packet.