We present a study of the incremental projection method to solve incompressible unsteady Stokes equations based on a low degree non-conforming finite element approximation in space, with, in particular, a piecewise constant approximation for the pressure. The numerical method falls in the class of algebraic projection methods. We provide an error analysis in the case of Dirichlet boundary conditions, which confirms that the splitting error is second-order in time. In addition, we show that pressure artificial boundary conditions are present in the discrete pressure elliptic operator, even if this operator is obtained by a splitting performed at the discrete level; however, these boundary conditions are imposed in the finite volume (weak) sense and the optimal order of approximation in space is still achieved, even for open boundary conditions.