Comprehensive muon-spin-rotation/relaxation (mu SR) and neutron powder-diffraction (NPD) studies supported via bulk measurements have been performed on the ordered double-perovskite Sr2YbRuO6 to investigate the nature of the magnetic ground state. Two sharp transitions at T-N1 similar to 42 K and T-N2 similar to 36 K have been observed in the static and dynamic magnetization measurements, coinciding with the heat-capacity data. In order to confirm the origin of the observed phase transitions and the magnetic ground state, microscopic evidences are presented here. An initial indication of long-range magnetic ordering comes from a sharp drop in the muon initial asymmetry and a peak in the relaxation rate near T-N1. NPD confirms that the magnetic ground state of Sr2YbRuO6 consists of an antiferromagnetic (AFM) structure with interpenetrating lattices of parallel Yb3+ and Ru5+ moments lying in the ab plane and adopting an A-type AFM structure. Intriguingly, a small but remarkable change is observed in the long-range ordering parameters at T-N2 confirming the presence of a weak spin reorientation (i.e., change in spin configuration) transition of Ru and Yb moments, as well as a change in the magnetic moment evolution of the Yb3+ spins at T-N2. The temperature-dependent behavior of the Yb3+ and Ru5+ moments suggests that the 4d electrons of Ru5+ play a dominating role in stabilizing the long-range-ordered magnetic ground state in the double-perovskite Sr2YbRuO6 whereas only the Yb3+ moments show an arrest at T-N2. The observed magnetic structure and the presence of a ferromagnetic interaction between Ru and Yb ions are explained with use of the Goodenough-Kanamori-Anderson rules. Possible reasons for the presence of the second magnetic phase transition and of a compensation point in the magnetization data are linked to competing mechanisms of magnetic anisotropy.