From looking through the AMR file specfication at RFC-4867 Section 5 (and taking a glance at OpenCORE-AMR), the AMR file format does not store a timestamp of any kind.
In fact, the format is extremely simple. #!AMR\n (or some variation thereof) followed by speech frames.
So the answer is no, there is no such property. However, if you've still got the phone, I'd look more into interacting with the filesystem on it, that's probably storing a timestamp of some kind.