I recommend stabilising the current website while you work out how to migrate to the latest version.
Stabilise the Current Website
Backup the current website using Akeeba Backup or similar.
Upgrade to Joomla v1.5.26 (if the site isn't already on the latest version in the series) and apply all the relevant EOL patches which you can find at https://docs.joomla.org/Security_hotfixes_for_Joomla_EOL_versions.
Compile a list of the currently installed third party extensions.
Remove any third party extensions that are not being used.
Update the remaining third party extensions to the latest versions.
Check that none of the currently installed third party extension versions are listed in the VEL at https://vel.joomla.org.
Set the hosting to the latest version of PHP that Joomla 1.5 will run on (e.g. PHP 5.6 is usually fine but you will probably need to suppress "deprecated" warning messages.)
Migrate to the Latest Version of Joomla
Check on the template developer website whether there is a version of the template for the latest version of Joomla. If not there may be something similar from the same developer or another developer where you can add a custom CSS file to make it look like the current template.
Using the list of third party extensions from the previous step, investigate which extensions have equivalent versions for the latest version of Joomla and find substitutes for any extensions that have been discontinued since Joomla 1.5.
Depending on how many users, menus, categories, articles, contacts, modules, links and so on you have to migrate, decide whether to use a migration utility such as SP Upgrade or similar or whether it is practical to migrate items manually.
It is usually best to create the new website in a subdirectory e.g. /new until you are ready to make the new website live. This means you can take your time and test the new website thoroughly while the old website is still available to website visitors.
When the new website is ready, I recommend moving the current website into a subdirectory e.g. /old and the new website into the root folder so you still have the old website to refer to for a few weeks in case anything has been missed in the migration. (Remember to update the paths to the tmp and logs folders in Global Configuration).
Once the new website has been active for a few weeks and everything is working well, remember to delete the old version and you can then set the hosting to an up to date version of PHP.