Does this operator commute with the Hamiltonian operator?

[tarkin]

Homework Statement

Show that the mean value of a time-independent operator over an
energy eigenstate is constant in time.

Homework Equations

Ehrenfest theorem

The Attempt at a Solution

I get most of it, I'm just wondering how to say/show that this operator will commute with the Hamiltonian operator.
Is it correct to say that because they both act on energy eigenstates, they must commute?
If so, is there a way to show this?

 

[blue_leaf77]

What is the form of the time-dependent energy eigenfunction? Say at $t=0$ this eigenfunction is $|n\rangle$ having energy $E_n$, how does it look like at a later time $t$?