Can't access apple.com webservers, App Store, nor Software Update
There are two things I would try in this situation:


Flush the DNS cache. This can be accomplished by running the command dscacheutil -flushcache. After running this command, attempt to run the dig and ping commands again. If they return the expected results, you should be good to go. 
Verify valid and responding DNS servers are responding to DNS queries. Open System Preferences -> Network. Unlock the lock icon in the lower left corner if it is locked. Select the network interface you are currently using from the left column and click the Advanced button. Click the DNS tab.

Typically in the DNS tab, you will want to go with what ever is recommended by your ISP. If everything appears OK, consider swapping out the server addresses with one or more of the addresses from the list below:

Google DNS

8.8.8.8

8.8.4.4

OpenDNS (Not recommended if the device hosts an email Server)

208.67.222.222

208.67.220.220




Edit to add the following:

Another possible solution: Have you power cycled the networking equipment between your computer and your network connection? I have encountered inconsistent DNS lookups couple times with my Airport. Power cycling has resolved the troubles in at least one of those situations.