Is the speed of sound almost as high as the speed of light in neutron stars?

The neutron star crust is a solid and there are indeed elastic waves for which the speed of sound is controlled by the shear modulus. I'm not sure where you got your estimate of the shear modulus from (there is some literature on the subject, see for example http://arxiv.org/abs/1104.0173).

Most of the neutron star is a liquid, and the speed of sound is given by the usual hydrodynamic result $$c_s^2=\left(\frac{\partial P}{\partial\rho}\right)_{s}$$ In dilute, weakly interacting neutron matter the speed of sound (in units of the speed of light $c$) is $$ c_s^2 = \frac{1}{3}\frac{k_F}{\sqrt{k_F^2+m^2}}$$ where the Fermi momentum $k_F$ is determined by the density, $$ n = \frac{k_F^3}{3\pi^2}$$ In the high density (relativistic) limit the speed of sound approaches $c/\sqrt{3}$. In the center of a neutron star you get quite close to this. Interaction change this result by factors of order one (indeed, recent observations of neutron star masses and radii suggest that the speed of sound near the center is close to $c$), but as an order of magnitude estimate these simple results are quite good.