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.

If you are saying goodbye to someone and telling them that you’ll see them again in an hour, it’s possible to use:

À dans une heure.

It’s a complete sentence that basically means “See you in an hour”. It’s the same construction that you used to say “À tout à l’heure” (“See you later”). In the same way you can use that either for a duration or a time:

À demain. — See you tomorrow.
À la semaine prochaine. / À dans une semaine. — See you next week.
À ce soir. — See you tonight.
À bientôt. — See you soon.

“Au revoir” can be seen as a special case of this, if you want: it’s the contraction of “À le revoir“, basically “See you when we see each other again”. Obviously it’s become a locution on its own, just like “Adieu” (“See you when we meet God”, ie. “when we’re both dead”).

Note also that in a lot of cases, one uses “se revoir” (see each other again) when telling someone you’ll see them in some time.

On se revoit dans trois jours. — See you in three days.

English: “I’ll see you in an hour”

French: “on se voit dans une heure”