Show a Category X’s custom post type on Category X archive page?

I’d suggest adding something like this to the top of the theme file or wherever you want this content to appear in the archive, category or whatever..

// Check if it's a category or taxonomy archive
if( is_category() || is_tax() ) {
    // Grab the queried data, slug, tax, etc..
    $queried = $wp_query->get_queried_object();
    // Check taxonomy and slug are set
    if( isset( $queried->taxonomy ) && isset( $queried->slug ) ) {
        // Look for a landing page post type with a slug that matches the current queried slug
        $landing_page = get_posts( 'name=" . $queried->slug . "&post_type=landing_page&posts_per_page=1&nopaging=1' );
        // If the result wasn't empty
        if( !empty( $landing_page ) ) {
            // Output the title and content using the same filters WP uses in the loop
            echo apply_filters( 'the_title', get_the_title( $landing_page->ID ) );
            echo apply_filters( 'the_content', get_the_content( $landing_page->ID ) );
        }
    }
}

This will should do what you want without interupting the main category query for the archive.

Hope that helps.