Having fitted an nth order polynomial to one data set, how can I now fit that exact curve to a second data set and obtain goodness-of-fit data?

After fitting your first data set, go to the second data set. From Prism's menu of nonlinear regression analyses, choose the appropriate form of the equation. Then click the "Constants..." button (ver 3) or the "Constraints" tab (ver 4) and fill in the values for all the parameters.

The resulting curve will be the same one that you obtained from the first analysis--since you have fixed all the parameters, Prism actually has no fitting to do. Nonetheless, it still reports sum-of-squares and Sy.x to quantify goodness-of-fit to the second data set.