I did the customer update section from outside the Magento directory like this,

    <?php
    require_once ("../../app/Mage.php");
    umask(0);
    Mage::app();
    Mage::app()->loadArea('frontend');
    Mage::getSingleton('core/session', array('name' => 'frontend'));
    $layout = Mage::getSingleton('core/layout');
    $pageTitle = "";
    $_SESSION[msg] = "";
    if (isset($_POST['updateCustomer'])) {
        $customerId = $_POST['customerId'];
        $customer_email = $_POST['customerEmail'];
        $customer_fname = $_POST['customerFirstname'];
        $customer_lname = $_POST['customerLastname'];
        $customer_password = $_POST['customerPassword'];
        $shipping_id = $_POST['shippingId'];
        $shippingParent = $_POST['shippingParent'];
        $street = $_POST['shippingStreet'];
        $city = $_POST['shippingCity'];
        $postal = $_POST['shippingPostcode'];
        $telephone = $_POST['shippingTelephone'];
        $region = $_POST['shippingRegion'];
        $country = 'IN';
        $customer = Mage::getModel('customer/customer')->load($customerId);
        $customer->setWebsiteId(Mage::app()->getWebsite()->getId());
        if ($customer->getId()) {
            $customer->setEmail($customer_email);
            $customer->setFirstname($customer_fname);
            $customer->setLastname($customer_lname);
            if ($customer_password != '') {
                $customer->setPassword($customer_password);
            }
            try {
                $customer->save();
                $customer->setConfirmation(null);
                $customer->save();
                $_SESSION['username'] = $customer_email;
                $_SESSION['firstname'] = $customer_fname;
                $_SESSION['first_order'] = "Y";
                $_SESSION['user_id'] = $customer->getId();
                $customer->sendNewAccountEmail();
            } catch (Exception $ex) {
                
            }
            $address = Mage::getModel("customer/address")->load($shipping_id);
            $address->setCustomerId($shippingParent)
                    ->setShippingId($shipping_id)
                    ->setFirstname($customer->getFirstname())
                    ->setMiddleName($customer->getMiddlename())
                    ->setLastname($customer->getLastname())
                    ->setCountryId($country)
                    ->setRegionId($region)
                    ->setPostcode($postal)
                    ->setCity($city)
                    ->setTelephone($telephone)
                    ->setStreet($street)
                    ->setIsDefaultBilling('1')
                    ->setIsDefaultShipping('1')
                    ->setSaveInAddressBook('1');
            try {
                $address->save();
                $address->setConfirmation(null);
                $address->save();
            } catch (Exception $e) {
                
            }
        } else {
            echo "User does not exist";
        }
    }
    
    if (Mage::getSingleton('customer/session')->isLoggedIn()) {
        $customer = Mage::getModel('customer/customer')->load(Mage::getSingleton('customer/session')->getId())->getData();
        $address = Mage::getModel("customer/address");
        $shipping = $address->load($customer['default_shipping'])->getData();
    }
    ?>
    <form action="" method="post" name="thisForm">
        <input type="hidden" name="customerId" id="customerId" value="<?= $customer['entity_id'] ?>" />
        <input type="text" name="customerEmail" id="customerEmail" value="<?= $customer['email'] ?>" />
        <input type="text" name="customerFirstname" id="customerFirstname" value="<?= $customer['firstname'] ?>" />
        <input type="text" name="customerLastname" id="customerLastname" value="<?= $customer['lastname'] ?>" />
        <input type="hidden" name="customerDefault_billing" id="customerDefault_billing" value="<?= $customer['default_billing'] ?>" />
        <input type="hidden" name="customerDefault_shipping" id="customerDefault_shipping" value="<?= $customer['default_shipping'] ?>" />
        <input type="hidden" name="shippingId" id="shippingId" value="<?= $shipping['entity_id'] ?>" />
        <input type="hidden" name="shippingParent" id="shippingParent" value="<?= $shipping['parent_id'] ?>" />
        <input type="hidden" name="shippingFirstname" id="shippingFirstname" value="<?= $shipping['firstname'] ?>" />
        <input type="hidden" name="shippingLastname" id="shippingLastname" value="<?= $shipping['lastname'] ?>" />
        <input type="text" name="shippingStreet" id="shippingStreet" value="<?= $shipping['street'] ?>" />
        <?php
        $collection = Mage::getModel('directory/region')->getResourceCollection()
                        ->addCountryFilter('IN')->load();
        ?>
        <select class="txtbox" name="shippingRegion" id="shippingRegion">
            <?php foreach ($collection as $region) { ?>
                <option value="<?= $region['region_id'] ?>" <?php if ($region['region_id'] == $shipping['region_id']) { ?> selected="selected" <?php } ?>><?= $region['name'] ?></option>
            <?php } ?>
        </select>
        <input type="text" name="shippingTelephone" id="shippingTelephone" value="<?= $shipping['telephone'] ?>" />
        <input type="text" name="shippingRegion" id="shippingRegion" value="<?= $shipping['region'] ?>" />
        <input type="text" name="shippingCity" id="shippingCity" value="<?= $shipping['city'] ?>" />
        <input type="text" name="shippingPostcode" id="shippingPostcode" value="<?= $shipping['postcode'] ?>" />
        <input type="text" name="shippingCountryId" id="shippingCountryId" value="<?= $shipping['country_id'] ?>" /> 
    </form>