Workup will depend on suspected etiology, may include
POC glucose
CBC
BMP
ESR
Lyme titer
RPR (if suspect syphilis)
Lumbar puncture to exclude meningitis, Guillain-Barré
Antinuclear antibody test
CT head: exclude acute bleed or mass
MRI: indicated for brainstem findings on the exam and to exclude pontine glioma in children and in adults where the abducens nerve palsy does not improve
POC glucose
CBC
BMP
ESR
Lyme titer
RPR (if suspect syphilis)
Lumbar puncture to exclude meningitis, Guillain-Barré
Antinuclear antibody test
CT head: exclude acute bleed or mass
MRI: indicated for brainstem findings on the exam and to exclude pontine glioma in children and in adults where the abducens nerve palsy does not improve
Sixth nerve palsy
Increased intracranial pressure 
Third nerve palsy
Vascular: Aneurysm, central venous sinus thrombosis
Fourth nerve palsy
Inflammatory: Post-viral, demyelinating, sarcoidosis, giant cell arteritis, Miller-Fisher Syndrome variant of Guillian-Barre Syndrome
Ulnar neuropathy
Neoplastic: Lesions in the cerebellopontine angle. In children – pontine glioma