I've been researching for our upcoming 4 day cruise. I have found out that the Rainforrest is $15/day, but you can buy a package. The rate of the package goes down each day you do not buy it. ie, $45 on day one, $40 on day 2, etc. I do not know if these are the prices for the package, but if you get a chance to see any navigators, the prices are listed there.
It does not appear to be private, tho there is a Private room you can sign up for - extra $$$$. Also look at a recent trip report, on the cruise trip reports, there is info in there too.
good luck