What is the best practice to style archive page 2,3,etc differently than archive.php?

In response to your comment, $wp_query->current_post in action. Note that it’s zero-indexed:

<?php

if (have_posts()) :
while (have_posts()) : the_post();

    if($wp_query->current_post==0):
        echo "this is the first post";
    else:
        echo "this is post number " . ($wp_query->current_post + 1);
    endif;

endwhile;
endif;

?>