Can rest and elevation help cure a sprain?

Yes. Rest, ice, and elevation are the best treatments for a sprain. Over the counter medications such as Ibuprofen may also be appropriate. An air cast and ACE wrap help to stabilize the joint and reduce swelling. Return to normal activities should be gradual.
Partly. Rest ice compression elevation are all standard treatments. You need to strengthen the injured part once it heals all the way.