Impossible to use the Package Center after upgrading DSM

Same thing happened to me. After upgrade it defaults to DHCP and changes it's IP address. Check your DHCP reservations and you'll find the device in the list. Connect to the web interface on that new IP and reconfigure it to use the DNS reserved IP you've been using in the past.

The problem has disappeared as soon as I installed DSM 5.2-5644 Update 3 a few days later.

So it really seems like an update-related issue, that's been fixed fast.

My situation was a bit different. I had installed the DNS package, but the DNS was disabled after an OS update. I had (stupidly) pointed the Synology configuration, under Network / General / Preferred DNS server, to the Synology DNS, not the external DNS. Since the Synology DNS was offline, it caused the connectivity problem (even though the network interfaces themselves pointed to the external DNS).

Changing the Preferred DNS back to the external DNS fixed the connection problem, and the package system started working again.