What attacks is it vulnerable to?
Denial of service comes to mind. E.g. someone sending data packets to the Uno at wire speed will probably be something of an issue to it. Something on the board needs to look into at least some part of the packets to determine if the software should care about it.
As far as I understand, the Wiznet chip on the Ethernet shield implements TCP and UDP itself, so it probably can filter out the uninteresting packets. But I can't find any numbers for what kind of performance you can expect. In any case, if the device is flooded with valid UDP packets aimed at the port your NTP client uses, the chip will likely need to pass them all to software. (If it supports filtering so as to accept only packets from the server you're trying to talk to, the attacker would need to spoof their IP address too.)
Depending on the specifics, the DoS could cause it to miss valid NTP replies, or just make it loop forever at looking at the network packets, without getting any useful work done.
Of course, you're probably not using any authentication with NTP, so forged/tampered reply packets could also be a possible issue.