The manifolds studied are almost contact complex Riemannian manifolds, known also as almost contact B-metric manifolds. They are equipped with a pair of pseudo-Riemannian metrics that are mutually associated to each other using an almost contact structure. Furthermore, the structural endomorphism acts as an anti-isometry for these metrics, called B-metrics, if its action is restricted to the contact distribution of the manifold. In this paper, some curvature properties of a special class of these manifolds, called Sasaki-like, are studied. Such a manifold is defined by the condition that its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). Each of the two B-metrics on the considered manifold is specialized here as an η-Ricci–Bourguignon almost soliton, where η is the contact form, i.e., has an additional curvature property such that the metric is a self-similar solution of a special intrinsic geometric flow. Almost solitons are generalizations of solitons because their defining condition uses functions rather than constants as coefficients. The introduced (almost) solitons are a generalization of some well-known (almost) solitons (such as those of Ricci, Schouten, and Einstein). The soliton potential is chosen to be collinear with the Reeb vector field and is therefore called vertical. The special case of the soliton potential being solenoidal (i.e., divergence-free) with respect to each of the B-metrics is also considered. The resulting manifolds equipped with the pair of associated η-Ricci–Bourguignon almost solitons are characterized geometrically. An example of arbitrary dimension is constructed and the properties obtained in the theoretical part are confirmed.