We investigate resonant localization and reflection-phase dynamics in graded phononic superlattices with spatially varying band edges. Starting from the elastic wave equation, we derive, using the local periodic approximation and the multiple-scale method, an effective Schrödinger-type equation for the envelope function, whose coefficients are expressed in terms of unit-cell averages involving the local Bloch mode. Spatial modulation of the local band edge then acts as an effective potential containing an internal turning point, allowing the system to be interpreted as an open quasi-bound scattering structure. For parabolic band-edge profiles, the effective potential becomes approximately harmonic and supports a series of quasi-bound resonances with a half-harmonic-like character. We show that the resonance frequencies form a nearly equally spaced series consistent with WKB quantization, and that the spatial mode profiles are accurately described by Hermite-function envelopes. The turning-point region is described by an Airy-function connection that consistently links the oscillatory and evanescent regions of the wave field. We further show that the semiclassical round-trip phase determines the quasi-bound resonance condition, while the Wigner reflection delay, defined from the total scattering phase, exhibits pronounced peaks near these resonances. These results establish a semiclassical description of resonant localization, quasi-bound scattering, and temporal delay in graded periodic wave systems.