Yes, assuming at least that the cone point is the only singular point. Hence any isomorphism will preserve the ideal of that point which is the ideal of elements of positive degree.(I think that the cone should in some sense be the most singular point in general and hence would still be preserved.) You can think of the isomorphism and its inverse as a graded isomorphism where the coordinate rings are graded by the powers of the ideal of the cone point. They then give graded isomorphisms of the associated graded rings. These associated graded rings are however the original coordinate rings. Hence we get a graded isomorphism of the coordinat rings and these isomorphisms are equal to those induced by the linear maps on the degree $1$ part.
Yes, assuming at least that the cone point is the only singular point. Hence any isomorphism will preserve the ideal of that point which is the ideal of elements of positive degree.(I think that the cone should in some sense be the most singular point in general and hence would still be preserved.) You can think of the isomorphism and its inverse as a graded isomorphism where the coordinate rings are graded by the powers of the ideal of the cone point. They then give graded isomorphisms of the associated graded rings. These associated graded rings are however the original coordinate rings. Hence we get a graded isomorphism of the coordinat rings and these isomorphisms are equal to those induced by the linear maps on the degree $1$ part.