I am not sure about determining whether an AOI in in the sun or shade at a certain time, however you can set the SUN to be on or off in the API by using
GESun.setVisibility
Edit:
Using the GE-plugin, create a LookAt with your desired AOI lat/long where the view is directly above looking straight down. Depending on the size of you actual AOI I would keep the view as low to the ground as possible.
Then capture a screenshot/image - I do not think this is possible through GE (if anyone knows a way I would like to find out), so maybe use javascript to take it - I found this Q on SO that provides some insight.
Take a screenshot with GESun.setVisibility set ON and then another with it OFF
Compare the two images for darkness/lightness or something and determine if your AOI is in the shade or not. You might find it better to surround your AOI in a Polygon of some sort in order to help your program distinguish it from the rest of the image - depending on the height the LookAt was taken from etc etc....
I do not have any ideas on how to compare the images, but yet again another search on SO resulted in this (I would presume finding the values of COLOR_BLACK in PHP ImageMagick) and this (Color Buckets idea).
Depending on your method of choice, it might help to alter your images to black/white before doing the comparing.