Make sure to stretch your hands before and after playing. Since the problem is also happening to you outside of osu!, I would consider stretching when you get up in the morning as well. Also, as a general rule, make sure to get enough sleep. Good sleep fixes many minor health problems.
When you start to feel this "tired" sensation, you should find a good stopping point and take a break from any strenuous hand activities for the rest of the day. If this problem does not improve while doing what I said, you should go to your doctor to get a professional opinion as this may be a more serious injury. Even if it is serious, there is a very high chance that your problem can be cured in a relatively quick amount of time.
I hope your hand gets better soon so that you can enjoy osu! again!
I actually used to have this exact issue (assuming it's the same issue) years ago. When I was at school my right hand would sometimes get so weak that I couldn't hold my pen anymore, let alone write anything down. Not sure what to say though lol.
But yeah, go consult a doctor instead, we're not exactly here to help you or even able to help you with issues like these.