We investigate the ground-state phase diagram of the quarter-filled Hubbard ladder with nearest-neighbor Coulomb repulsion V using the density matrix renormalization-group technique. The ground state is homogeneous at small V, a "checkerboard" charge-ordered insulator at large V and not too small on site Coulomb repulsion U, and is phase separated for moderate or large V and small U. The zero-temperature transition between the homogeneous and the charge-ordered phase is found to be second order. In both the homogeneous and the charge-ordered phases the existence of a spin gap mainly depends on the ratio of interchain to intrachain hopping. In the second part of the paper we construct an effective Hamiltonian for the spin degrees of freedom in the strong-coupling, charge-ordered regime that maps the system onto a frustrated spin chain. The opening of a spin gap is thus connected with spontaneous dimerization.