  In recent years, considerable interest has been drawn by the analysis of
geometric functionals for the excursion sets of random eigenfunctions on the
unit sphere (spherical harmonics). In this paper, we extend those results to
proper subsets of the sphere $\mathbb{S}^2$, i.e., spherical caps, focussing in
particular on the excursion area. Precisely, we show that the asymptotic
behavior of the excursion area is dominated by the so-called second-order chaos
component, and we exploit this result to establish a Quantitative Central Limit
Theorem, in the high energy limit. These results generalize analogous findings
for the full sphere; their proofs, however, requires more sophisticated
techniques, in particular a careful analysis (of some independent interest) for
smooth approximations of the indicator function for spherical caps subsets.
