Yes, "à une heure" does not mean "in an hour", but rather "at one".
You can't really ever use "à + moment" the way you're wanting to, at least with a verb in front of it. "À" pretty much always indicate something rather precise, like an actual time or a place. It really is the literal English "at". You could say "à l'heure venue" or something along those lines, but otherwise it's a no-go. If you want to use "à tout à l'heure", you have to say it as a farewell, all of it, since it's a fixed idiom. Otherwise you'll have to cut the "à":
Je te verrai tout à l'heure.
So how do you say "I'll see you in X time"? Well, your first attempt is actually what you'd say. It is the correct way to say it. It works with minutes, hours, days, weeks, months, etc. When talking to someone, I'm more inclined to use "on" and the present tense than "je" or "nous" and the future tense, but there's nothing wrong with it.
Je te verrai dans deux heures.
On se voit dans trois jours.
Nous nous verrons dans un mois.
In the case of "semaine", "mois" and "année", you can use "prochain(e)" instead.
Je te verrai la semaine prochaine.
On se voit le mois prochain.
Nous nous verrons l'année prochaine.
You can however say "à dans X time", without a verb, as Najib's answer very nicely explains.