Both "Have you managed to" and "Did you manage to" are fine in American English (AE). You are talking about a single past event.

Did you manage meaning?

to succeed in doing or dealing with something, especially something difficult: [ + to infinitive ] Did you manage to get any bread? I only just managed to finish on time. A small dog had somehow managed to survive the fire.

How did you manage to do?

The phrase itself (or any variant of it, such as, "How did you manage to do that?" or, "How did you pull that off?") expresses astonishment that someone was able to do something surprising. The question is essentially asking, "How did you do that?" but with an added sense of surprise.