Is there anyone who has experience of acquiring a second/additional language through exposure only, as an adult? Or language teachers or anyone else who knows about this process?

Basically, in my current WIP set 40,000 years ago, I have a Homo sapiens character who is exiled from his people and ends up living with a small clan of Neandertals. Neandertals would've had a similar language ability to modern humans, but their language would likely be extremely different (more different than any two modern languages) due to the populations being separated for much longer. None of the populations in my story have any kind of written language and I don't think their languages would've had different registers (formal, informal, etc) so it would be equivalent to only learning everyday informal language and not any academic/formal language. He's also completely immersed in it, entirely cut off from any Homo sapiens people.

He's been there long enough to have acquired some of the Neandertal language - enough to understand most of their conversations, but he doesn't find it easy to speak, partly because he's learned it as an adult and partly because their language is so different I think he'd struggle to pronounce some sounds and there'd be interference from his native language in terms of grammar errors. But I'm not 100% sure about the grammar errors, as I know that young people who acquire another language speak it with good grammar, but I don't know if that's true for anyone who learned the language entirely as an adult.

How long roughly would it take him to reach the kind of level I want him to be at for my story? I need to know this for the timeline of the backstory.

Also, how does the process work, is it a kind of linear progression where you understand a little more each day or do you go for a long time not really understanding anything then suddenly it all clicks?

Do people who acquire a language through exposure only as adults have good grammar? Do they go through a phase of speaking without good grammar on the way?

Is there anything else that would annoy you if I get it wrong?