Q: Update fails during download of Aperture. I have 10.8.3

You can try downloading it here.

Aperture

App Store Support

Where are you downloading it from - should have asked that question the last time.

Try setting up another admin user account to see if the same problem continues. If Back-to-My Mac is selected in System Preferences, the Guest account will not work. The intent is to see if it is specific to one account or a system wide problem. This account can be deleted later.

Isolating an issue by using another user account

If the problem is still there, try booting into the Safe Mode.  Shut down the computer and then power it back up. Immediately after hearing the startup chime, hold down the shift key and continue to hold it until the gray Apple icon and a progress bar appear. The boot up is significantly slower than normal. This will reset some caches, forces a directory check, and disables all startup and login items, among other things. If the system operates normally, there may be 3rd party applications which are causing a problem. Try deleting/disabling the third party applications after a restart. For each disable/delete, you will need to restart if you don't do them all at once.

Safe Mode

Safe Mode - About

General information.

Isolating issues in Mac OS X

Troubleshooting Permission Issues

Step by Step to Fix Your Mac

If you still can't get the App Store to work, I'd contact them via the App Store Support link.