Determine time given algorithmic complexity as input grows?
The standard method is to heuristically assume that a bound of the form $\Theta(g(n))$ can be approximated by $Cg(n)$ for some constant $C$, though in some cases this approximation is not justified, and the "constant" $C$ fluctuates (but is bounded on both sides).

Considering your first example, if $f(n) \approx Cn^3$ then $f(k^2) \approx Ck^6 = (Ck^3)^2/C \approx f(k)^2/C$. Unfortunately, the dependence on $C$ cannot be eliminated, but $C$ can be approximated empirically, though you should take such results with a grain of salt.