If you

Trace[NMinimize[x^4 - 3 x^2 - x, x]]

it will produce a stunning amount of output.

Peeking inside that are lots and lots of places where the method is being checked, used, and that look like they might be able to provide you with the method being used, like Optimization`NMinimizeDump`$DiagnosticLevel and many others.

Unfortunately that is long and complicated enough that I wasn't able to figure out how to get it to display the needed option value. Perhaps someone could take a moment to ask just the right person inside to answer this question.

Unfortunately AbsoluteOptions does not appear to have been uniformly implemented across all functions and so it isn't possible to easily and consistently get that information from all the functions that use Method->Automatic by default. That might be a really nice feature for the future and might only take a small block of almost standardized code inserted into each function.