If there were a simple proven way to do it, many computer science
students would jump for joy.  There isn't, though.  Generally speaking,
you take primitive operations (such as comparison or assignment) and
assign them some constant amount of time that they will take; then you
do some reasoning to find the smallest possible mathematical formula
that's always greater than the time required by the application.  Remove
the constant factors, and voila!  That's your big-O.

Of course the hard part is "do some reasoning to find [...]".  There is
intelligence required in applying the concept of big-O.