My arm itches for hours at a time. Is this an allergic reaction?

Possibly. Discuss this with your pcp and they might send you to an allergist for a more detailed evaluation. Questions that will be important relate to whether the reaction is localized to one arm and possible contacts to that area that are unique to it. You might also have eczema and a history and physical will help determine whether you need any testing or not.
Maybe. Itching is a common symptom of disease. The most common cause is dry skin. How to tell itching due to allergy from other causes of itching? Allergic rashes are always associated with a rash - hives, eczema, contact dermatitis. No rash? Think thyroid, hormonal, liver or kidney disease. But first try moisturizing with an unscented cream or ointment.