Updating ww.wp.xz.cn (beginner issues)

Hello all,

I’m a first-time user of ww.wp.xz.cn, I’ve always used wordpress.com. I recently started interning for this non-profit and they have a ww.wp.xz.cn website that someone else made a long time ago. I was updating the content of the website and I noticed that despite in my Dashboard the Menu elements would appear, they wouldn’t appear on the website itself. I figured it had something to do with the fact that I was still using 3.6.1 when on the Dashboard there’s a pressuring message regarding updating WordPress to 3.8.
Here’s my issue:
I tried to google backing up ww.wp.xz.cn and all the plug-ins require really advanced stuff that does not really work anyway (such as accessing your website database and reuploading through third-party softwares?). Anyway, I was wondering if by copying and pasting in a text file the website’s custom theme HTML code and then exporting the website contents was going to be enough as a backup.
I want to be able to recreate the same exact website if somehow the update goes terribly wrong.

Also, to update, do I need to know anything about the host log-in stuff. I did not create this website as I said… so I’m really not sure how to update the blog and having it backed up without screwing anything up.

Thank you,

Sara