Go by the expiration date printed on the package. For most OTC medicines, opening the container doesn't start a shorter countdown — storage is what actually shortens a medicine's life.
Yes — Robitussin (cough syrup) carries a printed expiration date and that's the date to go by. Use it by the expiration date printed on the bottle — for most OTC liquid cough syrups, opening does not start a short countdown the way a pharmacist-mixed antibiotic suspension does. Keep it tightly capped at room temperature. Not medical advice — follow the label and ask a pharmacist; don't take medicine that's expired or has changed in colour, smell, or texture.
Room temperature 68–77°F (20–25°C), tightly capped, away from heat and moisture — not the bathroom cabinet. Heat and humidity shorten a medicine's real life faster than the printed date suggests, which is why a bathroom cabinet is the worst place for it.
Throw it out if you notice: Discoloration; Changed or off smell; Separation or cloudiness that won't shake back in; Past the printed expiration date. Not medical advice — ask a pharmacist if you're unsure.