This does not answer the question, as it is still open, but there are some interesting related things to bring up.
First a group being $\delta$-hyperbolic is a coarse condition but being CAT(0) means that the group actually geometrically acts on a CAT(0) space―a space that is simply connected and has a metric non-positive curvature condition. If you take some hyperbolic group the question is: can I find a CAT(0) space the group acts on or how do I show there is no CAT(0) space the group acts on? There doesn't seem to be a nice universal construction, say coming from modifying Cayley graphs, which produces a CAT(0) space to act on and saying that a group doesn't act on a space, without "obvious" obstructions, is difficult.
Quasi-isometric to CAT(0) but not CAT(0)
Here is something that is known which should make you skeptical of an affirmative answer to the question: there are groups which are quasi-isometric to CAT(0) groups but are not CAT(0) themselves! So there are groups with no geometric action on a CAT(0) space even though it is quasi-isometric to a CAT(0) space. One class of examples come fundamental groups of graph manifolds. Leeb proved in 3-manifolds with(out) metrics of nonpositive curvature that there are graph manifolds without CAT(0) metrics. Then an arguement that this group has no geometric action on a CAT(0) space (I don't remember the paper but it is by Kapovich and Leeb). It is also known that all graph manifold groups are quasi-isometric to a graph manifold group which is CAT(0). In fact, it is known that all graph manifold groups are quasi-isometric (from graph manifolds without boundary).
Most of the obstructions along these lines essentially come from understanding high rank flats and using the flat torus theorem and other related theorems. These sort of arguments do not apply to hyperbolic groups since they do not have high rank flats/high rank abelian subgroups.
Random groups, depending on model, are hyperbolic and CAT(0).
There is no single definition of random finitely presented group but in most models that are studied random finitely presented groups are hyperbolic. In the paper Cubulating random groups at density 1/6, by Ollivier and Wise, they show that random groups (using density model with parameter 1/6) will also be CAT(0), in fact CAT(0) cubable. I am guessing there are other results along similar lines.