In each part of this question, list clinical examination findings for each of the two underlined conditions that would help you to distinguish between them:
Ulnar nerve injury
Lower brachial plexus
This forces the recall of medical school level neuroanatomy.
Particularly, the supply distribution of the brachial plexus
Ulnar nerve palsy would be associated with weakness of only the small muscles of the hand, and of sensory loss over the area supplied by its palmar sensory branch (essentially one-and-a-half fingers worth, over the palm).
A brachial plexus C8-T1 injury - also known as Klumpke's paralysis - would result in a "claw hand", with a supinated foream and flexed fingers. The sensory loss would include the whole ulnar nerve distribution - but also the medial cutaneous nerve of the foream.
Lastly, damage to the nerve roots at C8-T1 will also result in a Horners Syndrome, as the ascending sympathetic supply will be interrupted.
For a reference, I direct the time-rich reader to Sir Sydney Sunderland's "Nerves and Nerve Injuries", from 1968. (Not available as full text in Google, unfortunately.)
For brachial plexus injuries, I recommend Alain Gilbert's book.
For peripheral nerves, there is Haymaker, Webb, and Barnes Woodhall. Peripheral nerve injuries: principles of diagnosis. Thieme, 1998.