As long as the cavity is fully inside the metal sphere, it does not matter for this question whether it is off center. Metal being a conductor, it is not allowed to have any standing net electric fields inside of it. Conceptually, this means the electric field emanating from the +q charge inside the cavity must end on an equal amount of negative charge on the inner surface of the cavity inside sphere. Therefore, an excess charge of -q must reside on the surface of the cavity (all excess charge must reside on the surface in conductors).
The surface of the cavity cannot have *more* than -q on it, or there would be a net field inside the conductor. So the remaining -q excess charge on the sphere must migrate to the outer surface. Hence, the outer surface of the sphere must carry -q charge.
Note that, though the amount of charge on the outer surface is not affected by the cavity being off-center, its distribution will be. More of that outer surface charge will bunch up in the region farther from the cavity. If the cavity were on center, the outer surface charge would distribute evenly around the surface.