Earlier quoted context omitted.
IMAP "push" usually means the IDLE extension, where a client holds a connection open and waits for the server to report that something has happened. This works ok, but isn't great on mobile because holding a TCP connection open is usually difficult on flaky networks and consumes battery. iOS Mail doesn't implement it, instead doing a poll every 15 minutes. However it also implements a separate push system which allow…
So there's absolutely no way to enable that on custom little VPS? You have to make an agreement with Apple for that?
Disclosure: have family connection to owners.