People like proactivity and boldness. Indeed, this has been suggested to be why readers are often drawn to charismatic villains over heros, people are more drawn to the proactive, goal-oriented villains who often behave in a socially dominant manner and this is why readers will often white-wash or outright ignore the villainous actions of proactive villains while at the same time disliking heroic characters who aren't seen as assertive.
The protagonist of my urban fantasy story is not an assertive person. The best words to describe him would probably be neurotic, introverted, timid, and high-strung. His life has been thrust upside down into a world he didn't know existed (i.e., typical urban fantasy masquerade) and he's terrified out of his mind due to being trapped between the faction of monsters that wants to kill him and the faction of monsters who claim to be his friends, and he wasn't a very assertive person to begin with. He's not having much fun, he shows signs of PTSD and he would go home if it wasn't for the fact that he'd probably be killed if he left the safety of one of the supernatural factions. His entire story arc is about learning how to be braver and have more confidence in himself, ultimately culminating in him having a bit of self-actualization.
EDIT: Several respondents have pointed out that introverted and timid are not the same thing. This is true. The thing with this character is that they are both introverted and shy. Both character traits are often seen as negative by readers. People like extroverted characters because they like characters that are outgoing and social and extroverted characters have an easier time driving the story forward. People want to see interesting people doing interesting things, and if a character is reluctant to jump into the adventure they consider that an annoyance. Similarly, readers like seeing characters that are brave and exhibit socially dominant behaviors, for reasons that are probably too long to go into here. Hence, characters that are not outgoing and do not attack every problem head-on have an uphill battle to win reader sympathy if they are the protagonist. Side characters in general seem to get a bigger pass.
The problem with this is I'm worried about how to keep his character from coming off as whiny or annoying, given how the character naturally isn't very adventurous or gung-ho. This is especially the case at the beginning of the story where he is at the most controlled by his flaws, compared to later as he undergoes character development and starts being a bit more brave and assertive. People like characters who jump into the adventure and never look back, not ones who get dragged into it kicking and screaming, even if they have completely justified reasons to feel that way.
In general, the character is a deconstruction of the wish-fulfillment trope seen in a lot of urban fantasy where the protagonist becomes inducted into some kind of hidden world and manages to claw their way up to some position of social dominance like a vampire lord or a werewolf alpha (e.g., The Saga of Darren Shan, Kitty Norville, among others). The story goes out of its way to point out how utterly unsuited his is for this lifestyle, how just because he's become a supernatural he doesn't instantly skyrocket to social dominance because he's still the same dorky guy on the inside, and indeed how he is at a disadvantage compared to people who have fully acclimated or were born into this lifestyle because he lacks their innate viciousness from growing up in such a cutthroat world. Between this and the fact that his arc is about gaining self-confidence, it seems completely out of character for him to be assertive and adventurous. The problem with this is there's a reason why wish-fulfillment characters are a thing, people like the tropes even if they are completely at odds with reality.
He's also not in a position where he has a huge amount of power over events. He's a little fish in a big pond and part of his character at this point is that he's still trying to figure out his nature and as a newborn supernatural there are a lot of people that have a lot more knowledge or power than he does. He does try to effect things, but given his inexperience and lack of knowledge sometimes him taking actions either makes things worse or succeeds at resolving the problem with huge consequences. But the other characters mess up as much as he does, so it's not like he's a complete failure. This is a problem because when a character is introverted the solution the author comes up with to make them more sympathetic is giving them a goal for them to direct their attention towards (e.g., Frodo in Lord of the Rings), and people generally don't like characters who don't have the power to effect their surroundings.
He does have positive character traits. He's loyal, idealistic, highly moralistic, reliable, and responsible. Despite being constantly terrified he never explicitly shows cowardice, his actions being best described as a human Courage the Cowardly Dog. But as some have pointed out, those character traits are not as emotionally provocative or engaging as heroes that are brave, adventurous, etc.
Given all this, how do I keep this character from coming off as whiny and annoying before character development kicks in and tempers him a bit.