I think the continued reference to "180 degrees" and "360 degrees" here is confusing.
As I said, it's only really relevant to steady-state sinewaves.

If you want a pure time delay, just use a "shift register" approach - sample bits into the "register," then sample them out again at an appropriate rate to give the required delay.
The number of samples you need depends only on the resolution you want - it does not have to be just 180 or 360 or any such "magic number!"

If you want the delay to be a certain fraction of a cycle, then you will need to measure the period & calculate the sampling rate in real time - you'd probably want to fix the "register" size to cope with your "worst case"