This paper studies sparse ridge regression in the low-rank, high-dimensional setting that often arises when data collection is expensive. By exploiting the low-rank structure, we develop approximation algorithms within a relax-and-sparsify framework: solve a convex relaxation and then sparsify its solution to satisfy the original sparsity constraint. Specifically, we establish a tight rank-dependent sparsity bound for the convex relaxation by decomposing its support into two parts. This decomposition in turn motivates two approximation strategies, parameter reduction and binary redistribution, each exploiting a different part. Experiments on synthetic and real datasets show that the binary redistribution algorithms outperform the benchmarks in most instances.