How to automatically update ACF field value every year?

If you don’t want to use a scheduler (ex: Action Scheduler) or a cron job, and you only want to update once per year, then an option that tracks the next update time is probably the simplest approach (untested):

add_action( 'init', static function () {
    $option_name="cpt_ages_next_update";
    $option      = absint( get_option( $option_name ) );

    if ( $option > time() ) {
        return;
    }

    $query = new WP_Query( array(
        'post_type'      => 'cpt',
        'posts_per_page' => -1,
        'fields'         => 'ids',
        'no_found_rows'  => true,
    ) );

    if ( ! $query->have_posts() ) {
        return;
    }

    foreach ( $query->posts as $post_id ) {
        $year = absint( get_field( 'birth_year', $post_id ) );
        $age  = absint( date( 'Y' ) ) - $year;

        update_field( 'age', $age );
    }

    update_option( $option_name, time() + YEAR_IN_SECONDS );
} );

I would recommend using a plugin like Action Scheduler though, especially if you the number of posts is very high.