Write to Craft via Element API
I'd caution against trying to adapt the ElementAPI plugin for this purpose, as it was designed to be read-only—it's basically a wrapper around Craft's ElementQuery interface, which does not handle write operations.

In lieu of that, you might take a look at the CraftQL plugin, by Mark Huot. It supports sophisticated queries (reading), as well as mutations (writing).

Otherwise, if the scope of the content you need to update is relatively narrow (i.e. you know what section and fields you need to use), I'd recommend reading through the plugin docs, and determining whether it's something you could build on your own, with a simple Controller and supporting Service(s)!