I am having problem with updating an array element with in $_SESSION variable of PHP. This is the basic structure:
$product = array();
$product[
Don't blindly stuff the product into your session. Use the product's ID as the key, then it's trivial to find/manipulate that item in the cart:
$_SESSION['cart'] = array();
$_SESSION['cart'][$id] = array('type' => 'foo', 'quantity' => 42);
$_SESSION['cart'][$id]['quantity']++; // another of this item to the cart
unset($_SESSION['cart'][$id]); //remove the item from the cart