Upgrading from any version of Joomla 1.5.x to a newer one involves overwriting the outdated Joomla files. For this purpose Joomla provides upgrade patches for all of its previous versions to the latest one. You can find here the patch for your installation and make sure you download the one with .zip extension.
After you have downloaded the Patch Package to your local computer you should upload it to the installation directory of your Joomla.
If you have cPanel, you can upload and extract the package following the steps below:
Step 1: Open your cPanel -> File Manager and navigate to the installation folder of your Joomla application. Then click on the [Upload] button.
Step 2: Browse through your files and select the Patch Package from your local computer.
Step 3: Click on the Patch Package file in the File Manager. You need to extract the archive file using the Extract function. Click the Extract button and the files will be extracted upgrading your Joomla to the latest version.
If you don't have cPanel or you feel more comfortable using an FTP client, then you can simply extract the Joomla 1.5 patch package to your computer and then upload the extracted files to your Joomla folder using your favorite FTP client.
Overwrite the existing files with the new ones and your Joomla 1.5 will be upgraded.
In some cases the upgrade may corrupt additionally added components and modules on your Joomla. In this case you will have to re-install them.