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 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, possibly highly graded, simplicial meshes. The lifting includes a global coarse-level solve with the lowest polynomial degree one together with local contributions from the subsequent mesh levels. These contributions, of the highest polynomial degree $p$ on the finest mesh, are given as solutions of mutually independent local Dirichlet problems posed over overlapping patches of elements around vertices. The construction of this lifting can be seen as one geometric V-cycle multigrid step with zero pre- and one post-smoothing by (damped) additive Schwarz (block Jacobi). One particular feature of our approach is the optimal choice of the step-size generated from the algebraic residual lifting. Numerical tests are presented to illustrate the theoretical findings.