What is the viscosity of Sun surface? It should be possible to compute value from hydrodynamics of Solar prominence. What is the current estimate of Solar surface viscosity?
 A: 
What is the viscosity of Sun surface?

By "solar surface" I am going to assume you mean the photosphere.  In that case, then the question is whether you are asking about dynamic ($\mu$) or kinematic ($\nu$) viscosity.
According to a paper by Cowley [1990], the relevant parameters for the photospheric surface are: T ~ 4700 K, $\rho$ ~ 4.9 x 10-9 g/cm3, and ne ~ 1.1 x 1011 cm3.
We know that the following is an okay approximation $\mu \propto \rho \ \lambda \ V_{T}$, where $\lambda$ is the mean free path and $V_{T}$ is the thermal speed.  One can approximate $\lambda \propto (n_{e} \ \sigma)^{-1}$, where $\sigma$ is the collisional cross-section (here Coulomb collisions).
From Table 2 in Cowley [1990], we find that $\mu$ ~ 4.1 x 10-4 g cm-1 s-1 and since we know $\rho$ we can see that $\nu$ = $\mu/\rho$ ~ 8.4 x 104 cm2 s-1.

What is the current estimate of Solar surface viscosity?

I imagine things have not changed much other than more accurate density and temperature constraints for various regions.  However, any model that attempts to make things more accurate will quickly become nearly impossible without simulations (e.g., adding multiple ion species with energy-dependent cross-sections is not a trivial task when they have different temperatures and densities).
