Get better soon

Get better soon
 

Can we say improve yourself meaning the same?

No. That's a literal translation that doesn't work.

'Get better soon' is said to those who are sick.
It isn't used when you want someone to improve themselves, or make improvements.
You can get better at something or you can improve yourself, but not because you're sick. You can get well (soon), or get better (soon), only when you're sick.