I run Little Snitch to monitor my network traffic. However, on Mountain Lion, power nap runs while my computer is sleeping so I can't approve new rules for Little Snitch to allow power nap to connect to Apple and the App Store. What rules should I add to Little Snitch to allow power nap to operate properly?
Power Nap is not an application, it is a set of hardware-level functionality that causes the Mac to wake from sleep, but only with disk, CPU, and networking hardware active - no video or audio. However, third-party applications run as normal during this "DarkWake" period (Apple's term). Therefore, Little Snitch should use whatever rules you have in place while the computer is fully awake.
This appears to be confirmed by the mention of Power Nap on Little Snitch developer objective development's blog, in an article about Automatic Profile Switching. That functionality found in Little Snitch allows the system to automatically switch rules based on what network your Mac finds itself on. As long as you're on a network that you've previously used, and have rules set up for, Little Snitch should use those same rules during DarkWake. Presumably you allow all of the things that Apple would do during DarkWake when you're actively using your computer (checking email, App Store updates, Photo Stream updates, iCloud document sync, etc.) so the rules already in place should suffice for use during DarkWake.