The correct pronoun would be, Each king must have his castle;
Because the first one, Each king must have it's castle, does not make sense because of the It's, a king is a male person so it should be his.
It's not the third one because, Each king must have his or her castle, doesn't make sense because a king is a male not a female so her would not be used to define a man.
The fourth one would be incorrect too because, Each king must have their castle, makes no sense either because their is for more than one person and in this case we are just talking about kings no one else
Hope this helped! :)