Which all ports are used by Apple Push Notification service?

Push providers, iOS devices, and Mac computers are often behind firewalls. To send notifications, you will need to allow inbound and outbound TCP packets over port 2195. To reach the feedback service, you will need to allow inbound and outbound TCP packets over port 2196. Devices and computers connecting to the push service over Wi-Fi will need to allow inbound and outbound TCP packets over port 5223.

    

    The IP address range for the push service is subject to change; the expectation is that providers will connect by hostname rather than IP address. The push service uses a load balancing scheme that yields a different IP address for the same hostname. However, the entire 17.0.0.0/8 address block is assigned to Apple, so you can specify that range in your firewall rules.

    

We can Try running a telnet command on your server to see if the server can reach APNs, like this:

$ telnet 1-courier.push.apple.com 5223

$ telnet gateway.sandbox.push.apple.com 2195

$ telnet gateway.push.apple.com 2195

References:
https://developer.apple.com/library/ios/technotes/tn2265/_index.html#//apple_ref/doc/uid/DTS40010376-CH1-TNTAG41