The key thing to understand with level-up is that it is the “Eureka!” or “aha!” moment—the point when everything you have been trying, testing, studying, and practicing “all comes together” or “clicks” for the character, so that they can actually use it.
As the quotation marks might hint at, this concept isn’t exactly foreign to real life, at least as far as English-language idiom is concerned. We have phrases for this kind of thing because it’s something we experience, though it is rarely as clean as D&D level-up.
The other thing to keep in mind is that characters are doing a lot of things that don’t get explicitly mentioned during game time: they exercise, or practice, or care for equipment, or whatever else, and maybe it gets mentioned a little bit, but it is mostly in the background.
The two of these things combine in an interesting way: you can freely have a character level-up, gain some new ability, and then say that some parts of that ability were starting to be seen before the level-up, it just was one of those things that didn’t get mentioned ahead of time. Suppose, for example, the barbarian had not told you they were planning to follow the Path of the Totem Warrior: you could easily still say that they had been noticing the effects of this spirit before leveling up. That helps to explain it, and helps to make the game feel a little bit more “real.”
But since you do know, that’s even better: you can throw these little bits of detail into the game now, and it will make their level-up seem even more “real” when it happens. That’s great!
So yes, you have a great idea, and this works perfectly well
It is not the only good way you could do it. You could also have something more sudden happen in-character, a spirit suddenly showing up, or they go on a vision quest, or whatever, but this way is good. You have exactly the right idea, and timing it for 2nd level is perfectly correct. Congratulations on an excellent approach to this question, I think you are learning to DM very well.