Attach an handler to the change event for the text box that grabs its value, finds all the options that start with that, and remove and prepend them to the beginning of the select list like this:
$('#myTextBox').on('change', function(){
var $select = ;
var textBoxValue = $(this).val();
$('#mySelect option').filter(function(){
return this.innerHTML.indexOf(textBoxValue) == 0;
})
.remove()
.prependTo($('#mySelect'));
}