Lagrangians of the abelian Gauge Theory and its dual are related in terms of a shifted action. We show that in d = 4 constrained hamiltonian formulation of the shifted action yields hamiltonian description of the dual theory, without referring to its lagrangian. We apply this method, at the first order in the noncommutativity parameter theta, to the noncommutative U(1) gauge theory possessing spatial noncommutativity. Its dual theory is effectively a space-time noncommutative U(1) gauge theory. However, we obtain a hamiltonian formulation where time is commuting. Space-time noncommutative D3-brane worldvolume hamiltonian is derived as the dual of space noncommutative U(1) gauge theory. We show that a BPS like bound can be obtained and it is saturated for configurations which are the same with the ordinary D3-brane BIon and dyon solutions.