How to display information from a web API on an LCD (2x16) in real time?
Your API presents the information in a format called JSON, JavaScript Object Notation. It is a human readable way of passing around data structures. Web APIs often make use of it.

Benoît Baluchon has a library, Arduino JSON for manipulating JSON in embedded systems. You can download it from GitHub. The documentation is here. Assuming that the problem you need to solve is getting the information out of JSON and into a form that you can display, this looks like a good place to start.

Using Arduino JSON, the code to get the first time would look something like this:

char* getSchedule();

StaticJsonBuffer&lt;200&gt; jsonBuffer;

JsonObject&amp; root = jsonBuffer.parseObject(getSchedule());

char* next = root["response"]["schedules"][0]["message"];


Note: this is from a quick search on "arduino json" followed by quickly reading the code. I doubt if it is exactly right, but hopefully it is enough to get started.