How does one manage to "time stamp" their physics paper without submitting it to a preprint server like arXiv? Say, I have a paper written up. I wish to publish it in a reputed journal without submitting it to arXiv. The first journal(A) I submit it to rejects it.  I send the paper to a 2nd journal (B) for submission. At the same time, some other group submits a paper to another journal (C) and it gets published there. Is there any way to avoid this problem without using arXiv?
My main advice is that this is not something you need to do.
You could use an alternate repository.  Many universities have one.  Some have embargoes.
The old way to do it was to mail the paper to yourself in a sealed, postmarked envelope.  I suppose you could have it notarized as well.