In this work, we consider conforming finite element discretizations of arbitrary polynomial degree ${p \ge 1}$ of the Poisson problem. We propose a multilevel a posteriori estimator of the algebraic error. We prove that this estimator is reliable and efficient (represents a two-sided bound of the error), with a constant independent of the degree $p$. We next design a multilevel iterative algebraic solver from our estimator and we show that this solver contracts the algebraic error on each iteration by a factor bounded independently of $p$. Actually, we show that these two results are equivalent. The $p$-robustness results rely on the work of Schöberl et al. [IMA J. Numer. Anal., 28 (2008), pp. 1-24] for one given mesh. We combine this with the design of an algebraic residual lifting constructed over a hierarchy of nested unstructured simplicial meshes. This includes a global coarse-level lowest-order solve together with local contributions from the subsequent mesh levels. These contributions, highest-order on the finest mesh, are given as solutions of mutually independent Dirichlet problems posed over patches of elements around vertices. This residual lifting is the core of our a posteriori estimator. It also determines the descent direction for the next iteration of our multilevel solver, which we consider with optimal step size. Its construction can be seen as one geometric V-cycle multigrid step with zero pre- and one post-smoothing by (damped) additive Schwarz. Numerical tests are presented to illustrate the theoretical findings.