This paper addresses an online convex optimization problem where the cost function at each step depends on a history of past decisions (i.e., memory), and the decision maker has access to limited predictions of future cost values within a finite window. The goal is to design an algorithm that minimizes the dynamic regret against the optimal sequence of decisions in hindsight. To this end, we propose a novel predictive algorithm and establish strong theoretical guarantees for its performance. We show that the algorithm's dynamic regret decays exponentially with the length of the prediction window. Our algorithm comprises two general subroutines of independent interest. The first subroutine solves online convex optimization with memory and bandit feedback, achieving a $\sqrt{TV_T}$-dynamic regret, where $V_T$ measures the variation of the optimal decision sequence. The second is a zeroth-order method that attains a linear convergence rate for general convex optimization, matching the best achievable rate of first-order methods. The key to our algorithm is a novel truncated Gaussian smoothing technique when querying the decision points to obtain the predictions. We validate our theoretical results with numerical experiments.