Let $G$ be a connected graph of order $n$ with domination number $γ(G)$. Wang, Yan, Fang, Geng and Tian [Linear Algebra Appl. 607 (2020), 307-318] showed that for any Laplacian eigenvalue $λ$ of $G$ with multiplicity $m_G(λ)$, it holds that $γ(G)\leq n-m_G(λ)$. Using techniques from the theory of star sets, in this work we prove that the same bound holds when $λ$ is an arbitrary adjacency eigenvalue of a non-regular graph, and we characterize the cases of equality. Moreover, we show a result that gives a relationship between start sets and the $p$-domination number, and we apply it to extend the aforementioned spectral bound to the $p$-domination number using the adjacency and Laplacian eigenvalue multiplicities.