We know what "any time now" means, indeed: from now on it will happen soon. But in the sentence, which is quite informal, "Now yo'berths's ready any time, Miss" - said by the porter in the train (Pale Horse, Pale Rider, K.A. Porter) -, does it mean the same? I'm not sure, because the words are not in the same order.
In the example, the berth itself is ready and will remain so until Miss is ready to board it.
I interpret "any time now" as similar to "real soon now" in that both refer to things that have not yet happened, and different in that "any time now" implies something will (or is likely to) happen, while "real soon now" often is used in sense "not going to happen".
In the example, when the porter says "Now yo'berths's ready any time, Miss", he or she means that the berth is ready now, and Miss may occupy it whenever she likes ("any time").