We consider a general relativistic fermion antifermion pair that they interact via an attractive Coulomb type interparticle interaction potential in the 2 + 1 dimensional spacetime background spanned by cosmic string. By performing an exact solution of the corresponding fully-covariant two body Dirac Coulomb type equation we obtain an energy spectrum that depends on angular deficit parameter of the static cosmic string spacetime background for such a composite system. We arrive that the influence of cosmic string spacetime topology on the binding energy of Positronium-like atoms can be seen in all order of the coupling strength constant, even in the well-known non-relativistic binding energy term (proportional to alpha(2)(c)). We obtain that the angular deficit of the static cosmic string spacetime background causes a screening effect. For a predicted value of angular deficit parameter, alpha similar to 1 - 10(-6), we apply the obtained result to an ortho-positronium, which is an unstable atom formed by an electron and its antimatter counterpart a positron, and then we determine the shift in the ground state binding energy level as 27, 2 mu eV. We also arrive that, in principle, the shift in ground state binding energy of ortho-positronium can be measured even for alpha similar to 1 - 10(-11) value with current techniques in use today. Moreover, this also gives us an opportunity to determine the altered total annihilation energy transmitted by the annihilation photons. The yields also impose that the total lifetime of an ortho-positronium can be changed by the topological feature of static cosmic string spacetime background. In principle, we show that an ortho-positronium system has a potential to prove the existence of such a spacetime background. (C) 2020 The Author(s). Published by Elsevier B.V.