I maintain my Unix website on, presently, a Mac. Can I maintain it from a (not jailbroken) iPad 3?
Sounds like Diet Coda would be right up your alley. It's an iPad rethinking of the highly regarded Mac App Coda.

Basically it's everything you need to do hand coding of websites in one app package — a text editor for coding, (S)FTP for file management, and SSH for any server-side tasks. That's a pretty bare-bones description, there are a lot of well polished details that you may appreciate, I suggest checking out the site for more information.
You may also be interested in reviews from Macworld and Macstories.
Backups
It's worth noting that Diet Coda doesn't store the site resources on your iPad (it may cache some for performance reasons, but no permanent storage), all editing is done directly on the server. For this reason, it's probably best to have a backup of your website stored elsewhere if you don't already. This could be your Mac, but you could keep it all online by using something like Amazon S3. From the looks of your site (it appears to be just HTML and images, no big downloads, etc.), you could almost certainly keep a backup there for just over a dollar per year. It would be easy enough to set up an rsync script to automatically back up your site on a regular schedule to S3, and even keep snapshotted versions, you wouldn't really have to do anything beyond the initial set up.