Update database record in plugin

It’s really hard to understand what are you actually doing ( a refactor will help also).

but for updating you can try:
You will need the $id, show you tell which row to update.

$wpdb->update(
    $wpdb->prefix . 'tropix_dmin',
    array( 'post_name1' => $arr),
    array( 'id' => $id ),
    array( '%s' ),
    array( '%d' )
);