I have sprained my hand while doing push-ups I...

regarding your question
hand sprain means  tearing of the ligaments of the hand resulted from  overtwisting of the
hand
symptoms:
1.pain in the joint
2. swelling
3. bruises
4.decrease ROM "range of motion"
5. tender to touch
diagnosis
hand spain is diagnosed by
1. taking a  full medical history from you
2. doing a clinical examination
3. doing hand x ray
treatment
1.rest for at least 48-72 hr
2. apply ice pack every 20 minutes
3. applying an hand bandage to compress the
hand
4. elevate your hand to reduce the swelling '
5. take an analgesic drugs like ibuprofen 600 mg three times daily