I need to perform some task (log the event) at this moment. How can I do this?
You can download the Android source code, modify it to incorporate your desired security flaw, compile the results into a ROM mod, and install that ROM mod on your own device.
This is not possible via the Android SDK.
There is a catch. I want to identify these instances (moments) for other running apps (not for my app) (Ex: I want to know when DropBox app is about to access the internet.)
You are welcome to install and use a firewall package on your rooted Android device and perhaps hook something up that way, for this specific scenario. In general, what you seek is not possible, for blindingly obvious privacy and security reasons.