There are two important classes of surfaces in the hyperbolic space. One of class consists of extrinsic flat surfaces, which is an analogous notion to developable surfaces in the Euclidean space. Another class consists of horo-flat surfaces, which are given by one-parameter families of horocycles. We use the Legendrian dualities between hyperbolic space, de Sitter space and the lightcone in the Lorentz-Minkowski 4-space in order to study the geometry of flat surfaces defined along the singular set of a cuspidal edge in the hyperbolic space. Such flat surfaces can be considered as flat approximations of the cuspidal edge. We investigate the geometrical properties of a cuspidal edge in terms of the special properties of its flat approximations.