First, I should explain that arthritis can occur at any age, but “arthritis” is not simply one thing. There are many types of arthritis from juvenile arthritis to osteoarthritis. By far the most common is osteoarthritis, but rheumatoid arthritis is perhaps the form that creates the most fear. And moreover arthritis may come alone or be a part of another disease or syndrome. You are right that the age of 21 makes many types of arthritis uncommon. The fact that you spend a lot of time working at a computer makes other reasons for your wrist and hand pain more likely.
One of the most common causes of wrist and hand pain in young people that frequently use computers or keyboards is carpal tunnel syndrome. This is a repetitive use injury
that results in swelling in an area in the wrist where the nerves that supply the hand travel through. The most common symptoms are pain and numbness
or tingling in the wrist and fingers that are especially worse at night. The treatment is usually fairly simple and includes taking NSAIDs such ibuprofen, reducing the level of activity that is contributing to the problem (such as typing) and wearing wrist splints either just at night or during the day as well. Without more information I suspect this may be the cause of your symptoms, though other types of inflammatory arthritis such as rheumatoid arthritis are certainly possible. You can try the above remedies and in the meantime make an appointment to see your primary care physician
who can better evaluate your symptoms and determine the best course of treatment. Good luck!