I understand he's basically "You shouldn't just wait. Approach him on your own"
But what does the repeated noun here adds?
- Inoue, for your part, you should talk to him instead of just waiting around.
- Inoue, you have your own problem! Don't just wait, why don't you approach him on your own?
AはAで is an important construction that basically means "in A's own way". Here it's used to indicate that Inoue has his own problem (although the other person has a problem). You haven't provided any previous context, but a sentence like this never appears without context. There should be a criticism of someone else just before this, for example, something like "佐藤はもっと周囲に気を配り、仲間はずれの人がいないか気をつけるべきだ".
See the following questions for similar examples.
It's used when the process/reason/method is different but the essential result/conclusion is the same.
And the answer to this question lists several patterns that look similar.
In the case where N is animate noun, followed by a verb referring to behaviour, the meaning of NはNで is '(that is) just like N', 'N, typically enough':
文子は文子でなにも手伝ってくれなかった 'That's Fumiko for you, not a bit of help.'
指揮者は指揮者で給料もらわなきゃ食えないしね "For his part, the conductor has to get a salary or he can't eat, you see."
井上は井上で待つだけじゃなくて would normally mean 'Inoue, being Inoue, not only does he wait, [but]...'; is there some error in the text quoted?