Magento 1.9 Custom Attribute Value – Dropdown Input Type

I figured it by myself... extension attributes are for more complex info.

For this particular case where it is a "simple" custom attribute (of type string), the way for updating via API is just to pass it inside of custom attributes array:

    $headers = array(
        'Content-Type: application/json',
        'Authorization: Bearer ' . $token
    );
    $data = array(
        'product' => array(
            'custom_attributes' => array(
                '0' => array(
                    'attribute_code' => 'needs_sync',
                    'value' => '0'
                )
            )
        )
    );
    $data = json_encode($data);
    $requestUrl = $this->_baseUrl . '/index.php/rest/V1/products/' . $sku;
    $ch = curl_init($requestUrl);
    curl_setopt($ch, CURLOPT_CUSTOMREQUEST, "PUT");
    curl_setopt($ch, CURLOPT_HTTPHEADER, $headers);
    curl_setopt($ch, CURLOPT_POSTFIELDS, $data);
    curl_setopt($ch, CURLOPT_RETURNTRANSFER, true);
    $result = curl_exec($ch);

First get the product ID using Product Repository and Object manager

$objectManager = \Magento\Framework\App\ObjectManager::getInstance();

// Get the Product Repository
$productRepository = $objectManager->get('\Magento\Catalog\Model\ProductRepository');

// Get the Product Id using sku
$product = $productRepository->get($productSku);

// Create the product object using product ID
$product = $productRepository->getById( $product->getId());

// Call the Attributes 
$product->getData('product_weight_class');
or
$product->getAttributeText('product_weight_class');

Try this and update.