You could do something like the following (merely an example, do not take literally):
$posts = get_posts( 'posts_per_page=-1' );
foreach ( $posts as $post ) {
$url = url_encode( get_permalink( $post->ID ) );
$get = wp_remote_get( "https://www.google.com/analytics/api?visits_for_url=$url" );
if ( $data = wp_remote_retrieve_body( $get ) )
update_post_meta( $post->ID, 'analytics_view_count', $data );
}
This process would then run as a daily/half-daily WP cron event.
I have zero experience with the API, so how you query data by URL, or as to what data types are returned, you'd need to read up on the documentation.
I would also assume that it has a query limit - you might need to throttle the process, or at least break it up into batches (perhaps the latter, especially if you have a large number of posts).