How can I programmatically backup Android phone's settings?

If you know the location of the settings database, you can use LlamaLab's Automate to copy app data within the application itself, even in the root directory.

If your phone is an LG, it's easy to do that with LG backup. To backup your phone's settings, go to Settings - LG Backup [in list mode] - Backup. Unselect everything except settings, and click backup.

If not, try Titanium Backup.

The easiest way availaboe to you is using a third party app. I found My Backup Pro (there is a free version as well) which seemed promising.

You can also search he market for settings backup.

^{I use an system app provided by the manufacturer which is called backup and restore. I'll give them if the mentioned solution does not work for you. The reason im asking you to go with the first app is that the manufacturer's system app is highly unlikely to work properly ina device other than the one is originally made for.}