"... after 24 hours of mailbox activation" means that the activation is going on for 24 hours. That's probably not what you mean. It probably doesn't take 24 hours of work to activate a mailbox. I think what you mean is that the mailbox is activated, and then 24 hours later "Communicator" is active. In that case, the first sentence is correct. Mailbox activation is a one-time event taking a relatively short amount of time. Then 24 hours later Communicator is active.
I'm assuming here that "Communicator" is the proper name of some software product, and that this "thing" becomes active at a specific point in time. If that's not true, the sentence has other problems.