To monitor your API in RapidSpike you should set your API Domain as a new website and page in RapidSpike.
For example, to test https://api.mywebsite.com/my-resource, the configuration would like like this:
You can then set up an Uptime monitor for the page. There are a number of options for the HTTP method, including GET, POST, PUT and DELETE. Read on for more information about POST and PUT requests.