What should be done for sprained ankle that is not painful or any swelling (feels slightly "off")?

PT. Best to to see a pt to strengthen the muscles on the outside of the ankle to restore stability.
Many options. Is this an acute sprain or joint instability after a sprain from the past? Rest, ice, elevation are good for acute. I like to use quercetin or turmeric as well to reduce inflammation. Wear supportive shoes or a brace/wrap at first. A chronic joint instability might be improved with rehab exercises and perhaps prolotherapy.
How long. Even though you do not have pain or swelling, if the ankle feels unstable or abnormal when moving the ankle you should be evaluated by a physician. Most likely a course of therapeutic exercise and proprioception will improve the symptoms, however, fracture or loose body should be ruled out through a history, physical examination and possibly xrays.
Get it checked out. A sprained ankle is not to be taken lightly, as it can create quite a few problems for you later on if it doesn't heal properly. Please visit a podiatrist at your earliest convenience for a thorough evaluation.