Efficient channel state information at transmitter (CSIT) for frequency division duplex (FDD) massive MIMO can facilitate its backward compatibility with existing FDD cellular networks. To date, several CSIT estimation schemes have been proposed for FDD single-cell massive MIMO systems, but they fail to consider inter-cell-interference (ICI) and suffer from downlink pilot contamination in multi-cell scenario. To solve this problem, this paper proposes a compressive sensing (CS)-based CSIT estimation scheme to combat ICI in FDD multi-cell massive MIMO systems. Specifically, angle-domain massive MIMO channels exhibit the common sparsity over different subcarriers, and such sparsity is partially shared by adjacent users. By exploiting these sparsity properties, we design the pilot signal and the associated channel estimation algorithm under the framework of CS theory, where the channels associated with multiple adjacent BSs can be reliably estimated with low training overhead for downlink pilot decontamination. Simulation results verify the good downlink pilot decontamination performance of the proposed solution compared to its conventional counterparts in multi-cell FDD massive MIMO.