Is every action from an amenable group amenable on a unital $C^*$algebra?

Yes it is. It follows from Theorem 3.3 of [1] and the fact that the trivial action of an amenable group on $\mathbb{C}$ is amenable. More modern reference is [2] (in particular Section 4.3). [1] C. AnantharamanDelaroche. Systèmes dynamiques non commutatifs et moyennabilité. Math. Ann., 279(2):297–315, 1987. [2] N. P. Brown and N. Ozawa. C*algebras and finitedimensional approximations, volume 88 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2008. 

