If you mean
you've got quite a ways to go, you're maybe a quarter of the way in? If that? You can get a vague sense of how much of the game is left by looking at the world map and seeing how much of it is filled in.
The Puni stuff got added to the game with a recent update, so you probably just missed it. Try going outside the atelier and looking at your porch; the screenshots/videos I saw of the Puni feature all took place there. (I haven't actually tried this myself so if it turns out I'm wrong, apologies!)
Travel bottles: there's a hard level cap based on the number of bottles you've found. Assuming that you've found three bottles, though, I don't think this is the issue. I'm not totally sure about this but I think just synthesizing new environments in each of the bottles will help you reach higher levels; my guess is that the ingredient you use for a bottle is not the sole determinant of the level of the bottle, and that it may also be affected by the level of the bottle you're synthesizing in and/or the levels of the other bottles you own. But I haven't experimented much with this personally. If you just want to get the party quest over with, you could try just finding a good bottle password. Not 100% sure it'll work that way but it's worth trying.