The last surving man on the planet goes on the run from tribes of horny women, desperate to be impreginated. Responsibility is the last thing this homosexual manchild wants.

There are the bones of an early Woody Allen movie here, but we need an inciting incident and an action. e.g.:

When the Viagra virus lays waste to the world's men, the last survivor must protect himself from the tribes of horny women by cross-dressing.

This has potential to be a wacky comedy and as Richiev specified there is inherit conflict from the nature of the character and the situation he is in.

Best to specify an inciting incident and goal though how about:
After all men on Earth are gone but one homosexual he goes on the run from tribes of horny women, desperate to be impregnated desperate to find a safe heaven.

Hope this helps

Being homosexual with the responsibility of repopulating the earth is enough conflict. He's gay, but must have sex with women or the human race will go extinct. That is good conflict.

You don't have to add that he's someone who runs away from responsibility, that takes away from the main conflict. He runs away because he's gay, not because he is irresponsible. That's just my thoughts anyway.