What I assume is that the mass is a property of the whole object and not just its surface. Keep in mind that you can also (for example) get many different 3D slices of the 4D object. Whatever the mass or its hyperdimensional equivalent would be, it should be a property of the whole thing and not some slice thereof. Still, a definite answer is not possible before a physical model is established.
Another argument against using the surface for calculating any properties like mass is the following: take a given volume of some material. You can form it in many different ways with almost arbitrary surface area (larger than a ball of the same volume). Would a change of the surface area change the mass? No, so the surface is not a good indicator for mass. My intuition says that it would also not work well in higher dimensions.
Sounds pretty awesome! Have you ever heard of Miegakure (sadly not available yet as far as I know)? Or 4D Toys from the same developer, which is actually finished and obtainable? There are also some games that are 3D but use non-Euclidean geometry (for example Hyperbolica).