How to show the last and newest modified post in a custom category?

I think the easiest way to get the newest modified posts on top of the loop is via post injection using the the_posts filter

add_filter( 'the_posts', function ( $posts, \WP_Query $q ) 
{
    // Only target the main query on the home page
    if (    $q->is_main_query()
         && $q->is_home()
         && !$q->is_paged() // Only target page one 
    ) {
        // Get the newest modified post
        $args = [
            'posts_per_page' => 1,
            'category_name'  => 'blog', // Note, this must be slug
            'order'          => 'ASC',
            'orderby'        => 'modified'
        ];
        $newest = get_posts( $args );

        // Make sure we have a post to inject
        if ( !$newest )
            return $posts;

        // We have a post, append it to our array of posts
       $posts = array_merge( $newest, $posts );
    }
    return $posts;
}, 10, 2 );

You can adjust this to your needs