The main problem is that you have a small detector one meter away from the source. This means most neutrons in the problem do not hit the detector and leave the problem with the CPU time wasted. Your spectrum is basic because only one neutron hit it.
I would consider calculating a slightly different problem that might be expected to have a similar result that can be calculated with less CPU time. Such as making the detector a ring, or even a spherical shell around the problem.
You could just run the problem for a long time. With 10^5 neutrons the spectrum has one or two neutrons. 10^7 a spectrum starts to appear. 10^9 would probably take 2 core days with MCNP5, and maybe twice that with MCNPX.
Or you could do both and get the spectrum 'right' with a bigger detector and then do a long run for the final result.
I note the X manual section H gives different constants for the watt spectrum of Cf-242, I do not know if the difference matters. I would also consider setting every bin of your energy range to match your real results.