I think given the fact that you are only 29 and it sounds like you are otherwise healthy that arthritis is low on the list of possibilities. However, if you have actual joint redness or swelling, arthritis might be a possibility, and you would need to see your doctor
about that right away.
Tendonitis in the wrist is very common, and it produces exactly the kinds of symptoms that you are experiencing. Tendonitis tends to be an over use injury, occurring in those who work repetitively with their hands (for example, typing at a keyboard). Treating tendonitis in this setting does require paying attention to ergonomics (getting a wrist pad for your keyboard for example), otherwise the injury will just keep recurring. Other than this, most acute cases of tendonitis will subside with ice, rest, and standard anti inflammatory medications like ibuprofen, which you can get over the counter.
If you have numbness or tingling in the fingers of your hand, that would point more towards carpal tunnel syndrome
rather than tendonitis which, regardless, is treated exactly the same way initially.
Please see your primary care doctor
to get a diagnosis and establish a treatment plan. Good luck.