1. Correctness of Grammar

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"

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.

1. Correctness of Grammar

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"

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 will improve at basketball. = I will improve (myself) 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.