Ok first off it sounds like you're really young and don't quite understand how insurance works, so I'll teach you a valuable lesson in life:
Insurance is just that, insurance that your product, whatever it may be, is covered that IN THE EVENT something bad happens you can get it replaced. Insurance is not something you buy AFTER an accident occurs, it is something you have BEFORE the accident so that you are covered. If you walk into the Verizon store and say "hey my screen is cracked, and I don't have insurance but I'd like to buy insurance so I can get this phone replaced for less than 1/5th the price" they will likely laugh at you and say sorry you're **** out of luck.
Now if you call up Verizon and say "hey I've been using this phone and I feel its fragile. I would like to purchase insurance for it. Can you please add the insurion insurance to my plan please?" I'm sure they'll gladly throw it on no questions asked. After its on wait a solid amount of time, as James said MINIMUM 24 hours then go into the store and show them your phone, tell them you have insurance and you need to get it replaced.
To be honest even 24 hours seems to close. I'd wait a solid week just to be on the safe side. They may not have any hard way to prove it happened before you got insurance, but you have no hard way to prove it didn't either.
The whole 30 day thing matters because when you buy a phone from Verizon you get 30 days to decide whether you want to purchase insurance or not. After the 30 days its basically too late. So I suggest you hurry up and call them. Now.
Edit: for the record I do not endorse this type of behavior. This IS insurance fraud. Anyone who thinks otherwise just doesn't want to accept it. I never have and never will do something like this. Just saying for the OP's sake, he needs to understand how these things work and since I'm such a nice guy, this method (even though illegal) is the best way about handling his situation.