From skimming the notes it looks like it's essentially a way to send APNS notifications to Mail.app and requires a cert from Apple. You could run it from your VPS once you get the cert.
I guess that this is achieved using a server-side communication: the Fastmail server send the notification to the Apple servers and the Apple server sends the notification to the user's device. So wireshark probably won't help.
From skimming the notes it looks like it's essentially a way to send APNS notifications to Mail.app and requires a cert from Apple. You could run it from your VPS once you get the cert.