Such a big update is pretty huge. The differences between Magento 1.3 and 1.9 are huge. I think upgrading your shop isn't the way to go. At least, it wouldn't be my way to do it. First of all, a lot has changed in the core code, but also in the templates, etc. So I think it might be better to download Magento 126.96.36.199, make a copy of your database, link the new Magento installation to that database copy and run the upgrades. You might want to set the default template in your database first, so the frontend will keep on working.
Since you say you know SSH, you might want to run the upgrade scripts using n98-magerun. Those work better then the default Magento commands, in my opinion.
If this seems to work, you can try to copy your custom theme to the new Magento installation. Setting that theme to be your default theme might result in errors (be sure to have developer mode enabled). So you'll have to fix these errors manually by making changes to your template files.
And last but not least, your webshop probably has extensions installed. These should be reinstalled as well. But the versions you're having in your 1.3 installation are heavily outdated. So you need to get the latest version of these extensions, remove the extension or replace it with a different extension.
All in all it will still be a lot of work to upgrade, from this version, but the key is that you take the time to upgrade everything properly as described above and step by step. Make sure that you have everything fixed before moving on to the next step. And if you're using Git, you might want to create a new branch there and commit every step you take.
Also, when something goes wrong, make sure you have a backup of your database available, so you can just reinstall the database and try again.