I took a quick look into Timmermann's book "An invitation to Quantum Groups".
It refers to:
Saad Baaj; Georges Skandalis Unitaires multiplicatifs et dualité pour les produits croisés de C*-algèbres Annales scientifiques de l'École Normale Supérieure (1993) Volume: 26, Issue: 4, page 425-488 ISSN: 0012-9593
They describe finite dimensional C${}^*$-Hopf algebras in terms of multiplicative unitaries. Theorem 2.2 shows that if you have commutative multiplicative unitaries you get a locally compact group and therefore in the finite case, a finite group.
This gives the first statement. The second statement is equivalent to the first by duality.
*edit*
Such a result is already stated as Theorem 3.3 in
L. I. Vaĭnerman and G. I. Kac, Nonunimodular Ring Groups and Hopf-Von Neumann algebras, Mathematics of the USSR-Sbornik, Volume 23, Number 2