Upgrading old CiviCRM/Drupal and switching hosts
I'd recommend something like this:


Take a snapshot of the filesystem and database(s) and get the current site running on the new host and new domain name. See: https://docs.civicrm.org/sysadmin/en/latest/misc/switch-servers/
Upgrade the filesystem to Drupal 7 (updating the CiviCRM package to CiviCRM 4.1.6 for Drupal 7 in the process of this upgrade so it is compatible but not functionality different beyond that).
Once Drupal 7 is running smoothly, you can upgrade CiviCRM to the most recent version. Backup databases often before attempting next steps.
Then, once CiviCRM is running smoothly on the latest version, you could either stay on Drupal 7 for the next few years OR upgrade to Backdrop CMS or Drupal 8 if you still have some energy.