How to update custom field of a posts in a particular category

Use get_posts to get all the posts in this category, then loop over it and update the post meta for every post.

<?php
//get_posts uses same parameters as WP_Query hence using 'category_name'
$tut_posts = get_posts(array('category_name'=>'tution', 'post_status'=>'publish'));
$url = //some youtube video url here
if(!empty($tut_posts)){
    foreach($tut_posts as $tut_post){
        update_post_meta($tut_post->ID, 'video', $url);
    }
}

Please check if this works for you.