I'm wondering exactly what you mean by "rotten"....if the wood is so soft that you can dig a finger into it, then the finished article is not going to be much use, so like Pete says, forget it. However, if the wood is reasonably dry and just has very soft/punky areas, there are things that you can do to turn it. There's a number of fluids that dry fairly rapidly and bind the wood fibres together in such a way that they can be turned. Cellulose sanding sealer, melemine lacquer, CA adhesive all work well to some degree. What I favour is a proprietary treatment for dry rot in window and door frames, companies like Cuprinol are the best to go for. Paint the fluid over the punky wood fibres and this stuff soaks in to several millimeters depth, saturating the fibres and then drying rapidly, hardening the fibres as it dries.It won't give you perfectly structured timber, but it does make an impressive difference.
As for which tools cut best on this sort of timber; I favour a freshly sharpened gouge with a light paring cut if possible (handle low down and using the wing of the gouge) on the outsides of bowls. On the inside of a bowl, it really is a case of trial and error!
Les