Wow. This reminds me of thingies I learnt for semiconductors, after the last ice age, so please take with mistrust.
The general idea is that heat excite the electrons from the ground state to the excited one. The electrons absorb energy for that. More internal energy at higher temperature means a heat capacity.
As a distribution, I'd take Fermi-Dirac. That's a long tradition for electrons.
Now, you have two electrons. You have to count the states : each one can be at ground energy or excite, and can be up or down, but if both are in the same state, they can't be both up nor both down. Fermi-Dirac then tells you to probability that multiplies the state count.
I have a problem with that, which the assignment hides, and doesn't provide data for. In the tiny NO molecule, I can't imagine that the orbital energies are independent of how many electrons are in that state and what their relative spin is. I believe the interaction is much bigger than 16meV. Whoever asked the question possibly ignores this. In an assignment, the best to do is to ignore completely this difficulty. In real life, it would spoil your model completely.
Then, you get an electronic energy from the occupation of the levels and their energy, as a function of the temperature. A derivate tells you the contribution to the heat capacity.
You "just" add the usual components of the heat capacity to that: molecular translations, rotations, no vibrations.
While vibrations are reasonably negligible for NO up to room temperature, by analogy with O2, the rotations at 1K are problematic, and I can't guess how detailed your answer is expected to be. At such a low temperature, the possible rotation states of the molecule have widely separated energies, so the contribution to the heat capacity isn't just 2*kT/2. I didn't check the detailed figures, but for H2 at 20K it makes a big difference, enough for a mix of ortho and para-hydrogen to evaporate without heat input, so I expect problems for NO at 1K. Maybe you're just expected to answer the wrong 2*kT/2.