For the purpose of computational efficiency, we propose two subspace-based methods, but without eigendecomposition, to address the two typical problems in nested array processing, i.e., direction-of-arrival (DOA) estimation and noise elimination. In detail, to estimate DOA parameters, we judiciously arrange the segments extracted from the co-array model and then introduce a novel co-array-based orthogonal propagator method (COPM). Next, we develop a projection-based noise cancellation approach in the co-array domain, improving the relatively poor performance of COPM at low signal-to-noise ratios. Simulations evaluate the proposed algorithms under both overdetermined and underdetermined conditions.