How can I harden the centre of a partly cooked christmas cake
I have never tried cooking a partially-done cake in a microwave to finish, but I don't expect it to turn out a desirable result.  You may be out of luck for the current batch.

In future, I recommend going by internal temperature, rather than total cooking time.  Depending on the recipe, cakes using wheat flour tend to set around 200F (93C), so aim for 205F; you may need to do a batch or two as an experiment before you find out the perfect "done" temperature for your cake.  Get yourself an instant-read thermometer or probe thermometer; make sure the tip of the probe is in the very middle of your cake when you take your readings.  Also, keep in mind that the cooking process continues even after you pull the cake out of the oven -- as the cake sits "cooling", the residual heat on the surface slowly penetrates to the middle.  Be sure to give your cake time to finish this resting process before you cut into it or stick it in the refrigerator/freezer.

As for hard-on-the-outside, there could be several reasons for this, but the one that comes to mind as likely is that the sides of the cake set too quickly for the recipe.  This could easily happen if the material of the pan you are using is too conductive.  Glass and ceramic bakeware heats slower than metal, and among metal cake pans, dark surfaces heat faster than dull or shiny ones.  If you are already using glass/ceramic bakeware, then get an oven thermometer to make sure your oven is heating to the correct temperature.  Even top quality oven thermostats rarely stay accurate over time.  If that doesn't reveal any problems, try dropping the oven temperature a little and extending the bake time.