Frequency dependent impedance calculation for microstrip
The skin depth doesn't really have much, if anything, to do with impedance. It simply defines a loss mechanism. Note that with a thicker track, you will have marginally less skin effect losses due to a larger surface area for the track.

The impedance of microstrip is inversely proportional to the thickness of the signal track, but the difference for 0.5 oz copper and 1 oz copper is quite small, and definitely smaller than the 10% impedance variation most fabricators will guarantee.

I often use surface layers for some high speed signals as there is marginally less dielectric absorption loss as the velocity of the signal is a bit quicker (due to the effective dielectric constant being lower as half of the signal is in air, not FR-4).

Surface layers are normally plated up to 1 oz or sometimes 2 oz thicknesses. Talk to your PCB supplier to find out what they normally do.

Your stackup looks normal for 1 oz plating on the surface and 0.5 oz plane copper thickness.