Hi,
Neither being a subdomain nor being on Cloudflare can cause a webserver to decide to emit an HTTP 503 code. To find out the server conditions at the time, you’d need to speak to the webserver admin so that he can investigate the logs. There’s literally nothing to look at in a plugin for that: as I say, the code’s defined meaning is that the webserver is telling you that it had a capacity problem at that moment.
You mention being on a VPS; in fact, if it’s an entry-level one, that actually makes instantaneous capacity problems more likely than shared hosting, because then you have a single CPU being shared across all the operating system, background resources, webserver, database server and PHP engine.
David