What medication should a nurse administer to a client exhibiting rigidity and drooling after fluphenazine administration?

The correct choice is benztropine because it is an anticholinergic medication that is commonly used to treat extrapyramidal symptoms, which can occur as adverse effects of antipsychotic medications like fluphenazine. Symptoms such as rigidity and drooling are indicative of acute dystonia or other forms of extrapyramidal reactions that may arise after the administration of typical antipsychotics.

Benztropine works by restoring the balance of neurotransmitters in the brain, specifically by antagonizing the effects of acetylcholine, thereby alleviating symptoms such as muscle stiffness and excessive salivation. This makes it an ideal choice for managing the adverse effects resulting from fluphenazine.

Other options like olanzapine and risperidone are atypical antipsychotics that might not provide immediate relief from the acute symptoms experienced. Propranolol, a beta-blocker, is generally used for cardiovascular issues or anxiety, and it does not address the specific neurologic symptoms induced by antipsychotic medications. Therefore, benztropine is the most appropriate and effective choice in this scenario.