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With the risk of sounding like a dork, the problem is that your temp directory is not writable. This is exactly what the error message tells you. Please note a common misconception here. Just the permissions alone do not a directory writeable make. It is the combination of both ownership and permissions which determines the writability of a directory. This is why the error message suggests that you either modify the permissions (fixing the permissions side of the equation) or enable FTP (fixing the ownership side of the equation). There is nothing else you can do. This is how the operating system works. I cannot do something against the way the operating system works. Neither does Joomla!.

I just realised you are using Joomla! 1.5 and an extremely old release of our software. This cannot be updated automatically any more. Please download Akeeba Backup Professional 3.4.3 from our Download page and install it on your site without uninstalling your existing version. This will perform the upgrade. From that point, please ignore all update messages; we stopped supporting Joomla! 1.5 beginning with Akeeba Backup 3.5.0.