Compton scattering
Main study material for this topic is the textbook by Vladimir Karas,
in particular, Sections 10.3 through 10.5. You may also check my
handnotes for some derivations.
Please, concentrate on the following issues:

coordinate system: the classical Compton scattering formula is derived in the frame in
which the electron is at rest before the scattering. Only in that
particular frame, the photon always looses its energy, while the
electron gains it. The inverse Compton scattering is not necessarily a completely reverse
process (i.e., the electron does not need to be at rest after the scattering), but
in general it refers to an event during which the photon gains energy in the oberver's
rest frame.

please, don't get confused by the notation; quantities
${\gamma}_{\mathsf{x}}$
refer to the electron, while
${e}_{\mathsf{x}}$
are energies of the photon

you should be able to derive the classical scattering formula, i.e., energy
of the photon after the scattering,
${e}_{\mathsf{f}}$,
as a function of its initial energy and the scattering angle in the rest frame of the
electron before the scattering

derivation of the energy of photon within the scope of the inverse Compton scattering
basically means transformation of the classical formula to arbitrary "observer's"
coordinate system. In practise, that system is not completely arbitrary, which comes
from the assumptions that are used throughout the derivation: the electron moves at
relativistic speed in the observer's frame,
${\gamma}_{\mathsf{i}}^{\mathsf{L}}\gg 1$, while the energy of photon before the scattering is much less than
the restmass energy of electron,
${e}_{\mathsf{i}}^{\mathsf{L}}\phantom{\rule{0.3em}{0ex}}{\gamma}_{\mathsf{i}}^{\mathsf{L}}\ll 1$

Formula for energy of the scattered photon in the observer's frame in the form
of equation (10.30) in the textbook is obtained with some further assumptions. The most
important one perhaps is that while the radiation field is roughly isotropic in the observer's frame,
in the rest frame of the electron (either before or after scattering), most of the
photons are coming from a narrow angle from the direction opposite to the motion of the electron
(as a consequence of aberation)

you are supposed to understand the idea of ZeldovichSunyaev effect, i.e. what are the incoming photons,
what is the scattering medium and for which astrophysical measurements this effect
can be used for. Note, that the key idea of the inverse Compton scattering holds also for
${\gamma}_{\mathsf{i}}^{\mathsf{L}}<1$, which is the case here (the intergalactic medium is kinematically hot, but the
electrons do not move with relativistic velocities)
While key aspect of the topic is to understand the difference between the Compton
and inverse Compton scattering, some other subtopics are considerd less important
during the exam, but may be of your professional interest

I will not ask you to write the KleinNishina formula for the Thompson scattering
cross section, though it would be good if you are able to sketch roughly its shape

the formula for net energy transfer due to inverse Compton scattering is usually not
a subject of examination

detailed derivation of the ZeldovichSunyaev effect presented in the textbook is
actually rather straightforward and you are encouraged to go through it. On the
other hand, it relies on the Kompaneets' equation, derivation of which we will skip
(interested students may search for it, e.g., in the textbook "Theoretical
Astrophysics" by T. Padmanabhan)