How long would it take for someone to show signs and symptoms of an air embolism from a syringe?

Immediately: Air embolisms of significant consequence are acute and often severe events clinically. Dysfunctional symptoms and signs appear essentially instantaneously if looked for. Small amounts of air such as small bubbles are usually of no consequence.

See below: Small amounts of air, less than 0.3 cc may be clinically insignificant. A larger bolus of air injected, probably greater than 2cc, could be significant, and symptoms/signs would probably be immediate. These could include chest pain or discomfort, cough, shortness of breath, decreased blood pressure, etc.