The first sentence is definitely wrong. The second one is better.

Here's a better rephrasing of the whole sentence:

My left eye was operated on a month ago.

The first sentence is definitely wrong, as concatenating the two phrases "I was operated on / my left eye" doesn't scan correctly.

The second one is better, as the preposition forms a transition between the phrases, but it's still longer than necessary to convey the intended meaning.

Here's a better rephrasing of the whole sentence:

My left eye was operated on a month ago.

Alternatively:

I was operated on a month ago, in my left eye.