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The shape of the Earth is mostly a result of its gravity pulling its mass into the most efficient shape available.  The oceans are also pulled towards the centre of mass, and collect on the lowest parts of the surface - covering a huge area to a depth of only a few miles because the planet is so smooth. 

If Earth managed to be sufficiently rigid and hold a cube shape, its gravity would still pull _on average_ towards its centre of mass, which would be the centre of the cube.  Any water would pool in the middle of the flat faces of the cube to form roundish oceans. 

The ocean(s) would bulge outwards as if trying to form parts of a sphere.  The gravity of the corners and vertices of the planet would flatten the bulge a bit, but they'd be a lot deeper (and therefore narrower) than the current oceans.  Unless there was far more water available, they would never reach the vertices of the cube so a boat couldn't sail from one face to another.  

You could think of it as a spherical planet with several enormous, wide ridges reaching several hundred miles high.  As well as the water, the atmosphere would pool in the lower regions (middle of the cube face(s)), so five of the cube's faces are probably inaccessible to humans until they learn to make pressurised vehicles.

Your explorer might be able to determine the shape of the world by other means. Ancient Greek-style geometry could be used to prove that the sea curves and the land doesn't, but confirming that it's a cube rather than, for example, a square-based pyramid would have to wait until someone crossed a vertex.