1. Correctness of Grammar
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No, the grammar is incorrect. Your suggestion of "I should improve myself to be a better person." is grammatically correct.

2. When to use a reflexive pronoun with "improve"
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If there is no explicit object, then the understood object is the person(s) indicated by the subject:

> I should improve. (no explicit object) = I should improve myself. (I =
> myself)
 
> They should improve (no explicit object). = They should improve themselves. (They = themselves)

> I want to improve at basketball. = I want (myself) to improve at basketball.

If there is an explicit object, and it's different from the person the subject indicates, then no reflexive pronoun is used:

> I should improve my score. 
>
> They should improve their efficiency.