How can I add dropdown widget/box to admin post page?

Since this is very similar to post formats (see post-formats) I would use a custom taxonomy.

That makes it easy to control the access level, and you get the meta box without writing extra code.

Then you insert the new post classes with a simple filter. Your theme must use the function post_class() – of course.

Example:

<?php # -*- coding: utf-8 -*-
/* Plugin Name: Post Class Taxonomy */

add_action( 'wp_loaded', 'register_post_class_taxonomy' );

function register_post_class_taxonomy()
{
    $caps = array(
        'manage_terms' => 'manage_options',
        'edit_terms'   => 'manage_options',
        'delete_terms' => 'manage_options',
        'assign_terms' => 'edit_others_posts',
    );

    $labels = array(
        'name'                       => 'Post Classes',
        'singular_name'              => 'Post Class',
        'search_items'               => 'Search Post Classes',
        'popular_items'              => 'Popular Post Classes',
        'all_items'                  => 'All Post Classes',
        'edit_item'                  => 'Edit Post Class',
        'view_item'                  => 'View Post Class',
        'update_item'                => 'Update Post Class',
        'add_new_item'               => 'Add New Post Class',
        'new_item_name'              => 'New Post Class',
        'separate_items_with_commas' => 'Separate Post Classes with commas',
        'add_or_remove_items'        => 'Add or remove Post Classes',
        'choose_from_most_used'      => 'Choose from the most used Post Classes',
    );
    $args = array (
        'rewrite'           => FALSE,
        'public'            => FALSE,
        'show_ui'           => TRUE,
        'labels'            => $labels,
        'capabilities'      => $caps,
        'show_in_nav_menus' => FALSE,
    );
    register_taxonomy( 'post_classes', 'post', $args );
}

add_filter( 'post_class', 'insert_custom_post_classes' );

function insert_custom_post_classes( $classes, $class="", $post_ID = NULL )
{
    NULL === $post_ID && $post_ID = get_the_ID();

    $post = get_post( $post_ID );

    if ( ! is_object_in_taxonomy( $post->post_type, 'post_classes' ) )
        return $classes;

    if ( ! $post_classes = get_the_terms( $post_ID, 'post_classes' ) )
        return $classes;

    foreach ( $post_classes as $post_class )
        if ( ! empty ( $post_class->slug ) )
            $classes[] = 'post-class-' . esc_attr( $post_class->slug );

    return $classes;
}