The patient presents with acute focal neurological deficits consistent with an ischemic stroke. The most critical initial step is to rapidly differentiate between ischemic and hemorrhagic stroke to guide treatment. Non-contrast CT head is the best immediate investigation as it is fast, widely available, and highly sensitive to detect intracerebral hemorrhage. This allows exclusion of hemorrhage before considering thrombolytic therapy. MRI, though more sensitive, is not typically first-line in the emergency setting due to time constraints. Carotid Doppler and nerve conduction studies are not urgent initial tests in acute stroke management.

Reference: Harrison's Principles of Internal Medicine, Jameson et al., 20th Edition.