Massive multi-input multi-output (MIMO) combined with orthogonal time frequency space (OTFS) modulation has emerged as a promising technique for high-mobility scenarios. However, its performance could be severely degraded due to channel aging caused by user mobility and high processing latency. In this paper, an integrated scheme of uplink (UL) channel estimation and downlink (DL) channel prediction is proposed to alleviate channel aging in time division duplex (TDD) massive MIMO-OTFS systems. Specifically, first, an iterative basis expansion model (BEM) based UL channel estimation scheme is proposed to accurately estimate UL channels with the aid of carefully designed OTFS frame pattern. Then a set of Slepian sequences are used to model the estimated UL channels, and the dynamic Slepian coefficients are fitted by a set of orthogonal polynomials. A channel predictor is derived to predict DL channels by iteratively extrapolating the Slepian coefficients. Simulation results verify that the proposed UL channel estimation and DL channel prediction schemes outperform the existing schemes in terms of normalized mean square error of channel estimation/prediction and DL spectral efficiency, with less pilot overhead.