Xrhrft

This question may sound silly, but it has been bugging me for years.

If I ask a question about a precise point in time, should I say "What time..." or "At what time..."?

For example,

At what time does Billy arrive at the swimming pool?

or

What time does Billy arrive at the swimming pool?

The initial preposition at in such contexts is entirely optional, but it usually wouldn't be included (although in reality we usually use when rather than [at] what time anyway :).

OP's specific example happens to include a "location-based" clause based on at [the swimming pool], but it might be worth looking at two slightly different contexts...

1a: What time does the shop open?

1b: At what time does the shop open?

1c: What time does the shop open at?

...and...

2a: Where did you come?

2b: From where did you come?

2c: Where did you come from?

In my opinion, both 'b' versions above are at least slightly stilted / awkward. But whereas 1a and 1c carry the same meaning, that's not the case with the second pair. 2a (without the preposition from) is effectively asking where you ended up, not where you started from (speaker might be asking your final position in a race, for example; Where did I come [in the marathon]? I did pretty good, actually - I came third, out of 2000 runners).

The point being that because there's no credible alternative meaning in the first pair that depends on whether the preposition is included or not, it's entirely a stylistic choice (and on average we don't bother with unnecessary words). But the second example shows that we do include the preposition where it's required to avoid ambiguity.

At what time.....? is the grammatically complete expression.
What time.......? is correct but more colloquial.

Note that the preposition “at” is often put at the end:

What time is your flight at?

Both the sentences are correct and mean the same thing. what time is just a shorter form of at what time

As user070221 notes, both sentences are commonly used in American English. In some formal speech and writing, "At what time" is more acceptable than "When" or "What time", especially when "a precise point in time" is being requested.

I am an American who grew up in a town with many native speakers of Spanish. To my ear, both examples in the original post sound like overly literal translations of "¿A qué hora?" To my ear, "When" is a more natural way of saying this in English:

When does Billy arrive at the swimming pool?