Stop WordPress Entirely and Run My Own Code

Question

You could hook init and check the current blog ID

function wpa85780_my_app_init(){
    $my_blog_id = 99;
    if( $my_blog_id == get_current_blog_id() ):
        include 'somefile';
        exit;
    endif;
}
add_action( 'init', 'wpa85780_my_app_init' );

This will exit before WordPress does the main query and loads the template.

EDIT

a slightly earlier action you could hook is after_setup_theme, that seems to be the earliest that get_current_blog_id will work. See the action reference page for the order of actions.