Oof. It's doable, but not easy.
First of all, you need a local web server with versions of PHP and MySQL contemporary to the Joomla 1.5 site you are trying to restore.
My strategy would be using a Windows 10 or 11 machine with a custom installation of Apache 2.4 (yes, it's newer than Joomla 1.5 but it's fine, this does not affect Joomla 1.5's operation!) , MySQL 5.0 and PHP 5.2.
The next problem to overcome is that the Kickstart versions from that time were hosted on JoomlaCode.org which shut down 2 years ago. So, instead, we have to use the very much obsolete Akeeba eXtract Wizard 3.3 to extract the backup archive.
The next problem you have to face is that you'd need to migrate that site to Joomla 4. Here's the documentation: Planning Migration - Joomla 1.5 to 4 - Joomla! Documentation
If you need old versions of our software, e.g. Akeeba Backup to carry out Joomla's official instructions, here's where you can find the historic versions: Compatibility
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!