In this article, we study approximation of solutions to a class of first order nonlinear neutral differential equations with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. Our main results are developed by utilizing analytic semigroup theory, fixed point theorem, and Gronwall`s inequality. Also, we study the Faedo-Galerkin approximate solutions and their convergence to the solution of our given problem. At last, an example involving a partial differential equation is presented to illustrate the discussed abstract results.