Encapsulation of client side logic in web page

Update: To answer your specific questions (which I just now realized I completely neglected):

1. The this/that stuff is one way of doing "context resolution" (you used the word "callback", but that's when you pass a function to another function). There are other ways, such as Function.bind/jQuery.proxy, but in your case assigning a local that variable is a valid and customary way of doing it.

2. Sure, you can have the constructor do nothing except declare and assign instance variables. But there are no written or unwritten rules regarding a construtor's responsibilities, so it's up to you what it should do. As you say the init method ends up being public, which isn't terribly useful - in fact, init is only useful once in the object's lifetime, so it should probably be in the constructor. And if it's in the constructor, you don't need to assign instance variables; they work fine as local variables and closures. And if they're closures, you don't need the this/that either...

... and so you don't actually need a "class" anymore, which is how I arrived at the code below.

It depends on your specific needs, but with the example you give, it seems like overkill to have a "class" (constructor + prototype) to handle that bit of logic. You can make do with:

$(function () { // shorthand for $(document).ready(...)
  var postNrCtrl   = $("#postnr"),
      postStedCtrl = $("#poststed");

  postNrCtrl.on("change", function () {
    var postnr = jQuery.trim(this.value);

    // if the field is empty, clear the postStedCtrl and stop
    if( !postnr ) {
      postStedCtrl.val("");
      return;
    }

    // Return if the value isn't 4 digits (i.e.
    // if it isn't a valid Norwegian postal code)
    if( !/^\d{4}$/.test(postnr) ) {
      // perhaps you want to alert the user here before returning
      return;
    }

    // Make an explicit JSONP request to get around same-origin policy
    // jQuery takes care of encoding the params and adding the callback param
    $.ajax({
      type:     "GET",
      dataType: "jsonp",
      url:      "http://fraktguide.bring.no/fraktguide/api/postalCode.json",
      data: {
        country: "no",
        pnr:     postnr
      }
    }).done(function (data) {
      if( data && data.valid ) {       // check the response a little
        postStedCtrl.val(data.result); // postStedCtrl is available via closure
      }
    });
  });
});

Here's a demo

On a more general note: For things like controller-logic, I find "classes" too cumbersome. You only need 1 instance of a controller, so coding a prototype and construtor only to instantiate it once, seems like overkill in a language like JavaScript that has object literals and closures.