Was there any kind of detective in Ancient Rome?

Probably not. Yes, the Romans had a kind of police called the "Vigiles", that was very different from modern police.As the name suggests, these were "vigilant" people, what we would call "night watchmen," whose job was to catch crimes in progress. This kind of "police" work did not need detectives.

The concept of detectives arose with the concept of modern police, which started in the late Middle Ages, probably in France, the richest, most crime-ridden country in western Europe. Here, the function of the police was not just to "stop" crimes in progress, but to suppress/root out crime. The first step was to interrogate/torture arrested criminals to find out who aided and abetted, or collaborated with them in their crimes. Later on, the police trained/hired detectives to cooperate with the police in gather evidence, or signs of criminal activity, while remaining functionally separate. This was so that crime could be "nipped in the bud" before it actually occurred.

But this prevention/anticipation function did not exist in Roman times, where police work consisted of "apprehension."

No, and yes. Mary Beard wrote an article for BBC History Extra describing crime in Rome.

The Vigiles were there to protect against fire - not to prevent, detect or deter crime. They might intervene in the case of crime if they felt obliged as a citizen, but their obligation was to stop things that were a danger to Rome, not to the inhabitants.

The case concerns a shop-keeper who kept his business open at night and left a lamp on the counter, which faced onto the street. A man came down the street and pinched the lamp, and the man in the shop went after him, and a brawl ensued. The thief was carrying a weapon – a piece of rope with a lump of metal at the end – and he coshed the shop-keeper, who retaliated and knocked out the eye of the thief.

This presented Roman lawyers with a tricky question: was the shopkeeper liable for the injury? In a debate that echoes some of our own dilemmas about how far a property owner should go in defending himself against a burglar, they decided that, as the thief had been armed with a nasty piece of metal and had struck the first blow, he had to take responsibility for the loss of his eye.

Note what is missing in this case - the shopkeeper never thought of summoning the police or taking legal action. He took direct, immediate action. Nor did the court consider that the thief was engaged in what we would today call a felony - they were only concerned with the loss of the eye. They had a very different notion of crime. (Bit of a paradox since they were (AFAIK) the first society to recognize the law as separate from the ruler. They displayed the law tablets, but never really used them.)

Here is a somewhat simplistic summary of criminal deterrence in Rome

The Law in Rome (Twelve Tables) - note that most of these crimes will be tried in civil court, not in criminal court (although the Roman's wouldn't have drawn the distinction, I think it is important for a modern understanding of Roman law.

Special military units enforced the law within Rome proper. The urban prefect (praefectus urbi) was a senator who commanded three cohorts (500 men each under Augustus, doubling to 1000 under Vitellius, and increasing to 1500 under Severus). These were responsible for policing ordinary crime in the city and within a 100-mile radius around it. CarolAshby

So why do I say that there were detectives? When all offenses are civil, you sometimes need to know who is stealing from your building site, or dumping toxic waste in your garden, or whatever. Patrons would have asked their clients to help in such situations, and clients would have hopped to to satisfy the demands of their patrons. They weren't professional detectives, but they were seeking the answers to mysteries on behalf of their "employers".

For example, Adultery was a crime - if your spouse was stepping out on you, your clients were obliged to protect your interests and might follow her to find proof. Inheritance law was big - if your siblings were carting off Dad's treasures while you were on military duties, your clients would probably track and report to you so that you could sue.

I can't point to any records, but I strongly suggest that the burial societies that formed the linchpin of middle/lower class social structures probably also made sure to look out for one another and protect one another's rights & privileges.

One could argue that at various points the Tribune of the Plebs was a sort of a detective - he was supposed to look out for the rights of the Plebs. But quickly the office was politicized.