Trouble updating a old wordpress site

hi. I have an old site I’ve inherited that is on version 3.5.2

Its causing some issues with my theme and plugins. When I try and do an automatic update it fails.

I’ve googled the issue and tried changing the permissions on the upgrade folder but it still won’t automatically work.

What can I do at this point? Is downloading the latest wordpress and uploading via ftp it to the server a good method? I’m not sure what else to do.

Here is my site: