Here's a trick. It isn't necessary to add javascript (or hook the colorbox close event) for this.
jquery.colorbox relocates your inline content into a structure it creates off the html>body root, before displaying it, and moves it back when it's closed. That's a strange behavior IMO, but take advantage and apply your 'hide rule' accordingly.
<style>
#div_id_i_want_to_show { display: block; ...your other style rules... }
#divParent #div_id_i_want_to_show { display: none; }
<style>
<div id='parent'><div id='div_id_i_want_to_show'>...
Alternately, invert the rules and place a 'reveal rule' that depends on a colorbox-defined element.
<style>
#div_id_i_want_to_show { display: none; ...your other style rules... }
.colorbox #div_id_i_want_to_show { display: block; }
<style>
<div id='div_id_i_want_to_show'>...
Oh, also, yet another option is to keep your #div_id_i_want_to_show inside a .hiddenDiv wrapper.
<style>
#div_id_i_want_to_show { ...your style rules... }
.hiddenDiv { display: none; }
<style>
<div class='.hiddenDiv'><div id='div_id_i_want_to_show'>...