The question is delicate because of the phrasing that assumes empirical approach, which did not emerge until 17th century. And the notion of "speed" as applied to sound presupposes the concept of wave in a medium, which dates to late Renaissance. The use of "speed" in antiquity was mostly confined to uniform motions of localized objects, not light or sound.
But the idea that sound takes time to travel is ancient if not prehistoric. The earliest surviving written record is perhaps Aristotle's Meteorology (c.334 BC), where he gives his theory of thunder and lightning: "...the dry exhalation that gets trapped when the air is in the process of colliding is forcibly ejected as the clouds condense and in its course strikes the surrounding clouds, and the noise caused by the impact is what we call thunder... the ejected wind burns with a fine and gentle fire and it is then what we call lightning, which occurs when the falling wind appears to us to be coloured. Lightning is produced after the impact and so later than thunder, but it appears to us to precede it because we see the flash before we hear the noise." (boldface mine)
General theory of sound is described in Aristotle's De Anima, but he mostly elaborates on what was a consensus among Greek natural philosophers since early pre-Socratics: sound is some kind of movement of air (with each part "striking the air which is next to it"), and no movement can be instantaneous because the same thing can not be two places at once. This was suggested by experience with noisy falls and hits, vibrating strings and air filled pipes in musical instruments, etc. The collision theory of thunder relies on such intuitions, and goes back to Anaximander (c. 611-547 BC) and Anaximenes (c. 585-528 BC). According to Aristotle, Empedocles (c.450 BC) even thought of light as "traveling or being at any given moment between the earth and its envelope, its movement being unobservable to us". On this account Aristotle disagrees because he considers light a state rather than a movement. But on sound there was complete agreement, I am not aware of anyone who thought that sound spreads instantaneously.
However, nothing resembling the expression "finite speed" was used anywhere in antiquity to describe either light or sound. Leonardo is credited with suggesting c.1500 that sound travels in waves in the modern sense ("we see the waves running across the field while the grain remains in place"). This allows one to make sense of speed and frequency. However, first measurements of the speed of sound were only attempted in 1630s by Mersenne and Gassendi, the first "authoritative" one was the one performed by the Florentine Academy in 1660.
