I'm not sure but I think a regular chest xray would show any kind of a problem, or if your doctor listened to your lungs he could tell if there was a problem. I think that you would have to aspirate a lot more than a piece of rice in order to cause an infection or something of that nature.
If you aspirated a piece of rice, it may take a long time for it to be absorbed. You might try to use gravity in your favor by coughing while in the head-down position. Rice is pretty bland and unlikely to cause infection. If you had aspirated something really irritating, such as a hot chili pepper, there would be more urgency to get it out. The over-the-counter cough expectorant might ease your cough but won't clear the rice. Neither will the antacid.
You should see your doctor and get a chest x-ray if you