A time multipoint nonlocal problem for a Schrödinger equation driven by a cylindrical $Q$-Wiener process is presented. The initial value depends on a finite number of future values. Existence and uniqueness of a solution formulated as a mild solution is obtained. A single-step implicit Euler-Maruyama difference scheme, a Rothe-Maryuama scheme, is suggested as a numerical solution. Convergence rate for the solution of the difference scheme is established. The theoretical statements for the solution of this difference scheme is supported by a numerical example.