It could be done with an automator workflow, for scheduling you could attach the workflow to a calendar event.
The workflow would be something like Get Specified Server -> Connect to Server -> Get Specified Finder Items -> Copy Finder Items
You could use the Put command to upload all the files that make up a website, but if only a few files have changed since the last upload, it isn't necessary or efficient to upload the entire site again. Instead, you can use the Mirror command to upload only new and changed files.
When you choose the Mirror command, Fetch displays the Mirror window, where you specify the source and destination folders to mirror between. They can be one folder on your computer and one folder on a server, or two server folders.
You can also use the Mirror command to completely synchronize two folders, deleting files in the destination folder that do not exist in the source folder in addition to copying new and changed files. You should use this feature with caution; see the Mirror window help topic for more information.
You can save the settings from the Mirror window as a mirror document, and simply open that document to restore the settings or start mirroring automatically.
For more information about mirroring and the Mirror window, see the introductory or in-depth Mirror window help topics, or the mirroring a local folder to a remote folder topic.