  The Holstein model, which describes purely local coupling of an itinerant
excitation (electron, hole, exciton) with zero-dimensional (dispersionless)
phonons, represents the paradigm for short-range excitation-phonon
interactions. It is demonstrated here how spectral properties of small Holstein
polarons -- heavily phonon-dressed quasiparticles, formed in the
strong-coupling regime of the Holstein model -- can be extracted from an analog
quantum simulator of this model. This simulator, which is meant to operate in
the dispersive regime of circuit quantum electrodynamics, has the form of an
array of capacitively coupled superconducting transmon qubits and microwave
resonators, the latter being subject to a weak external driving. The magnitude
of $XY$-type coupling between adjacent qubits in this system can be tuned
through an external flux threading the SQUID loops between those qubits; this
translates into an {\em in-situ} flux-tunable hopping amplitude of a fictitious
itinerant spinless-fermion excitation, allowing one to access all the relevant
physical regimes of the Holstein model. By employing the kernel-polynomial
method, based on expanding dynamical response functions in Chebyshev
polynomials of the first kind and their recurrence relation, the relevant
single-particle momentum-frequency resolved spectral function of this system is
computed here for a broad range of parameter values. To complement the
evaluation of the spectral function, it is also explained how -- by making use
of the many-body version of the Ramsey interference protocol -- this
dynamical-response function can be measured in the envisioned analog simulator.
