I think it was probably whenever Professor Sprout retired. That could have been anywhere from two year after the final battle, but was more likely closer to ten - thirteen. However, I don't believe that it was Albus' first year was Neville's first year as, by the way that James was talking, it seemed like he was well acquainted with Neville, not just as a family friend, but also as a Professor. He seems well established in the status of a Professor, so that he probably had held it for a long time before James came - maybe three years or so before.
When do you think Neville likely took this job?
I believe that Professor Sprout (who was likely the fondest of Neville when he was a student) would have probably recommended him, especially if he continued doing something with Herbology. He might have been a well established Herbologist, and I doubt after all he did during the final battle, that his name was something that would be easier forgotten among those had Hogwarts, so he would have probably been considered and offered the job. I don't really see him asking for it, though, for the simple fact that he always seemed to be very humble, and he would have felt honored when they considered him.
Would he have asked for the post or do you think the Head (McGonagall?) offered it to him?
Yes, I think he would want to.. It seemed to be the one thing he truly loved and excelled at. It was his passion. However, I do think that he might have dabbled in something else for a bit - like training to be an Auror. I think that after all he did during the battle, his grandmother would have pressured him to follow in his parents' footsteps. He might have gone along with it for a while, thinking that maybe that was something he should do, but eventually (maybe after someone gave him some advise or he was offered the job at Hogwarts [probably the first]) he'd realize that he'd had quite enough of fighting and he would much rather follow his passion. After he confronted his grandmother, she would probably be understanding, would want him to do what made him happy--after all that he's done, I don't think she could stop being proud of him.
Do you think his pre-Hogwarts Professor job would involve herbology in some way, given his interest for the subject?
I don't think that his weakness would be anything physical. In the Ministry of Magic, we saw his gumption, his ability to fight teeth and nails, and yet he managed to be captured by the Inquisitorial Squad. But I always recall the line that one of the Inquisitorial Squad said upon bringing him in: "I caught this one [Neville] trying to help the Weasley girl." (being a HUGE Neville/Ginny shipper, I love that line.) And that, in itself is where his weakness lies. He has a huge heart, and his loyalty to others - though an extremely honorable trait - would be his weakness. All his enemies would have to do is attack one of his friends or his loved ones and he would protect them, even at the cost of himself.
What could likely be Neville's weaknesses?
Also, I think that same desire to care for those he loves would lead him to be more timid. In his desire to make others happy, he would compromise himself, like the scenario I gave you about his grandmother and the Auror. However, I think this is something he would conquer or would only effect him slightly. But, yes, that's what his weakness would be.
Hope that helped, Akay.