The quickest thing I can come up with is by using an Arduino microcontroller with a Wemo switch.
Basically the Wemo switch will turn on & off the power to the Arduino. On the Wemo, you can set it to turn itself off after a certain amount of time. For this project, 1 minute auto-off would seem appropriate.
Then you'd write a simple program for the Arduino to close a switch when it first gets power. The program would open the switch 1 second later and then just sit idle. There are numerous pins available on the Arduino, just pick 2.
A minute later, Wemo turns off the Arduino.
Echo will not work without an Internet connection, mainly because it won't understand what you say to it. Anything you say after the wake word "Alexa" is sent to be processed at Amazon. The result is sent back to your Echo. This is how Echo, GoogleNow, Siri, and Cortana get far better voice recognition accuracy than non-cloud based systems.