All kind of headphones on my phone (Galaxy 4.1) won't work. I've come to learn that the problem isn't my headphones, but the outlet its self. Any way to fix this?

It seems to be a hardware issue, yeah? Sorry to say, there's no way of really fixing it. Do you still have the receipt, or some kind of warranty? I'd take it in and get a replacement.

Here's another option if that is not available: Blue tooth headphones/earbuds. The port for the earbuds may be damaged, but your phone has bluetooth. If it doesn't work still, then there's some kind of software issue.