Start with the basics: a main character in a predicament. Most teasers start with the existing flaw or handicap that hampers the character from addressing the main challenge of the film. In the case of Godzilla, show Tokyo vulnerable to potential monster attack or the chaos on the ship when the unexpected monster appears.
Show the tension leading up to the main conflict. In Godzilla, this would include lots of shots of people on the phone, getting into tanks, running for the door or looking up and screaming.
Include an authoritative voice-over to tie it all together. Pick whichever friend of yours sounds the most like that voice-over guy on tape. Write down all the lines and have them practice until it sounds professional.