Given that the bulk plasma in the device should not cool between each pulse below ~800C or so, you could just extend the helium gas cooling needed for the anode to the rest of the walls (making them out of something like tungsten), and run it through a brayton cycle turbine.
But as has been said before it would be much better to avoid having to use a thermal cycle and just extract even just 10-15% directly from the xrays to push over the Q=1 and dump the rest as waste heat.
asymmetric_implosion: The 500kV pinch voltage does not relate to the energy of the x-rays. The x-ray energy peak from bremsstrahlung is a function of electron temperature (http://en.wikipedia.org/wiki/File:Bremsstrahlung_power2.svg ). If the electrons are at ~150keV then the bremsstrahlung peak will be around a quarter of this – so around 30-40keV.