MC states he manages to get by in this world as a postman and has good reputation because he's not curious. Aka minding his own business. Get stuff done and leaves that's it. He is smart in that sense since he knows how this new world works and how to survive.

But in future following chapters he does the exact opposite and decides to eat a roasted monster tentacle he killed to protect himself to satisfy his random urges. (Like noncurious guy would not even think of eating strange dead monster. Usually curious ones are triggered to fulfill appeals to needs and objectives)

He also decides it's great idea to pretend to be the stranger guy he just killed in self defense, to avoid being imprisoned, when (1) he got rid of the evidence by burning the body; (2) he seem to forget he could just run away from the damn thing and ppl can assume the stranger guy could be killed by a monster instead.... and (3) he doesn't know a thing about this stranger (besides he has relations with monsters and tried to kill MC... and stranger lives in area MC is somewhat familiar with as he delivers to this area) and logically ppl can't suddenly be good at pretending to be someone else without knowing them to certain extent.

So in order for MC to get to know this dead stranger, he snoops and investigates around stranger's trap-filled house (aka another thing that is opposite to MCs said noncurious personality) to learn more about him and prepare himself to live there.

Author suddenly disregards this whole 'MC gotta pretend' situation and throws MC a mission where he has a different role and setup, and is suddenly transported to the mission area. Like I felt like the first situation wasn't resolved.... then well who cares MC u don't have to pretend to be dead guy anyway; u gotta do this instead... Cuz you should learn about your new system and ability.
Sure the whole u got sudden abrupt mission is not new but it just feels like the author just wanted force MC to be curious enough to chance upon this mission. If MC stayed true to his noncurious personality this wouldn't happened cuz he would've just avoided the place.

On this mission, MC is to be a nightwatchman, with his only goal to survive on Cross Street which has monsters.. And so because he doesn't know what's going on.. He ask questions about the whole scenario to clarify the whole situation, showing off how curious he is. //facepalm

Basically I wish the author didn't state the MC was not curious because it conflicts so much with future actions. If he only said MC was cautious then ok. There would be less glaring issue, but this is not the case. It's possibly one of the reasons I don't like noncurious MC's because author makes them do curiosity related things. As if it's forced and pretending 'oh MC is not curious he forced to be curious', but it's not. I sort of expect a noncurious MC to not be inquisitive. He could be more observant and figure out things without asking 'what is going on', but that is not the direction the author took.
And yes people could gradually be curious about things that interest them, but this is not the case.