how to retrieve date in computed field
It looks like you're indiscriminately popping values off of the array returned from field_get_items, without checking to see which index you actually need. I haven't tested this but I think you'll have better luck with code like this:

$items = field_get_items($entity_type, $entity, 'field_start_date');

// Use array_shift if you want the FIRST field item
$item = array_shift($items);
// OR
// Use array_pop if you want the LAST field item
$item = array_pop($items);

// Get the date value of the field
$start_date = $item['value'];

// Use start date
$entity_field[1]['value'] = $start_date;