No, it started when you read the series name for the first time. Level ups are something that happen in game worlds. I also prefer non-game interface worlds in isekai, but game interfaces are super popular both in Japan and Korea, so what can you do?
He ignored the first task the system gave him, and as a consequence was running for hours for his very life. Don't say he should keep ignoring stuff again. If the system gave him a private dungeon key, it's obviously for a reason, just like that physical training was. What others would he bring with him? He has got no buddies, as he's basically only baggage who's tolerated out of pity and general human decency, but that's it. It's entirely possible others wouldn't have even seen the entrance, nor be able to enter it, just like they didn't see the game interface.
So, all in all, his choices are to trust the system, that he can solve the current tasks with his own, still pitiful, strength or... die trying. In the end, if it looks like a game, it's not unreasonable to believe it's designed like a game. It's not like when you start a new game, it immediately throws you against the last boss, making it impossible for you to ever play further.