Is this scheme for a single-fan electric airship with a rocket-like geometry workable?
The best propeller driven vertical take-off and landing craft would be a helicopter, actually. Yes a duct around a propeller increases thrust, but upsizing a propeller creates way more thrust from the available power.

Propeller thrust has scaling effects with size due to different Reynold numbers, data does not scale up well. Best way forward from where you are now would be to look at the weight, rotor dimensions and installed power of existing designs for small helicopters such as the R-22. Or better yet, design an electrical version of the Hiller Flying Platform Platform.