This paper proposes a fast and decentralized solution methodology for the robust operation of multi-area integrated electricity-gas systems (M-IEGSs). A deterministic reformulation is obtained for the two-stage robust model by applying the linear decision rule based electrical reserve utilization scheme as well as regulating the distributed gas storages. Two linear approximations are developed for the nonconvex Weymouth equation in the gas network to determine the gas flow directions. The penalty convex-concave procedure (P-CCP) is then adopted to refine a feasible local optimum for the nonconvex model with an acceleration strategy. The decentralized decision-making is enabled by the alternating direction multipliers method (ADMM). The convergence as well as computation performance of the overall solution procedure can be guaranteed as only convex optimizations are solved. Simulation results validate the effectiveness of the proposed methods as well as the benefits of the proposed convex programing based solution procedure.