From 3D static order to 2D dynamic correlations in the tetragonal double perovskite Sr2CuWO6
\SCWO is a tetragonal ($I4/m$) double perovskite with diamagnetic W$^{6+}$ (5d$^0$) and (Jahn-Teller-active) Cu$^{2+}$ (3$d^9$) in a rocksalt-ordered arrangement on the B sites, modelling an anisotropic fcc-lattice antiferromagnet with $S=1/2$. We have studied the magnetic structure, dynamics and thermodynamic properties of this material above and below the \Neel{}-ordering transition at $T_{\textrm{N}} = 24$~K. Muon spin relaxation spectroscopy, neutron diffraction and neutron spectroscopy experiments show that the transition at $T_{\textrm{N}}$ on heating is from static 3D type-2 antiferromagnetism to 2D dynamic correlations that remain detectable up to 100~K. Above $T_{\textrm{N}}$ the muon relaxation is described by a compressed exponential decay, revealing a strong correlation between the spin fluctuation rate and the 2D correlation length. The low-temperature muon data and the entropy release around $T_{\textrm{N}}$ as obtained from the heat capacity data point to small magnetic domains, probably of the order of 60~Å. We note that the material behaves more like a quasi-2D square lattice antiferromagnet than could be expected considering its structure only. A possible explanation might lie in the unusual strong spin-orbit-coupled magnetism of the W 5$d$ levels, as the dominant exchange interaction is via the Cu-O-W-O-Cu pathway.