Hmmm. Definitely not my field, but this might be useful as a jumping-off point for people who know of older structures. I've been reading about Rome recently and immediately thought of the 128 AD Pantheon, but I believe it fails by two criteria here. It is definitely not the oldest manmade structure that's still in use today, and I think the ancient Romans would be very surprised to hear of its later use as a Catholic church and museum.

However, the Romans turned out to be a pretty good set of people to start with:

• Also in Rome is the Pons Fabricius, which is a Roman bridge built in 62 BC that connects the Isola Tiberina (or Tiber Island) to the Field of Mars/Jewish Quarter (or Campus Martius). The guy who commissioned it had his name engraved all over the damn thing so no one would be in any doubt that Lucius Fabricius was a civil servant par excellence who probably deserved a raise.
• However, after some nosing around, I believe the Turks might be able to lay claim to the title of the oldest continuously-used bridge (or at least, they've got a pretty good shot). The Caravan Bridge in Izmir was built in 850 BC and is still being used for traffic over the Meles river. However, finding quality information on this online has been tough.
• If you're looking for something a bit more active (not a useful way to describe it, I know, but I hope the distinction makes sense), the Proserpina Dam from the first to second century AD in Spain might be the oldest structure that's still doing most of what it was designed to do. It's a gravity-fed damn with sluices that used to feed the local aqueduct (no longer the case) and irrigate local farmers' fields (which it continues to do). By the way, your guess of Roman aqueducts was a good place to start, but frustratingly I can't find any that are still in use to supply drinking water. Irrigation, yes. One of our Roman experts might know.
• However, this one might be the closest to what you're looking for. The Dujiangyan dam/irrigation system near Chengdu, China was built in 256 BC, and continues to irrigate something like 3,300 square miles of land in addition to providing flood control. I'd read up on Chengdu's teahouse and relaxation culture previously, and was both surprised and not surprised to discover that this system might be one of the reasons behind its development. The story behind it and its architect, Li Bing, is actually so fascinating that I've made a note to go find a book on it later.

I don't know if any of these are the true oldest structures that are still being used for their original purpose, but the list here might be helpful to other commenters in constructing a more accurate account.

Would you consider the pyramids, and the many older tombs scattered across the world, as still fulfilling their original purpose after being excavated? If the purpose is "contain this body", then presumably not; if it is "memorialise this person", then at least those pyramids that can be attributed to a named pharaoh seem to be still "in use".

Here are past answers & discussions on similar topics:

Feel free to ask followup questions by commenting with /u/ appended to a flaired specialist's username (i.e. /u/OldWorldGlory).

Key:

c = comments
dy = days old
wk = weeks old
mo = months old
yr = years old

• [65 c | 2 yr] Flaired Respondents | Continual Use

• [39 c | 3 yr] Flaired Respondents | Currently in Use

• [19 c | 1 yr] Continual Use

• [14 c | 2 yr] Flaired Respondents | Currently in Use

Perhaps not what you are looking for, but roads are going to be the oldest structures. You run into a Theseus' ship problem of course, a road that has been patched up time after time can very well not contain its original elements anymore, and if maintenance became sporadic in less prosperous times, it may have been effectively a dirt road for part of its existence - while still being in continuous, albeit less intensive use throughout the times. Roman roads are the obvious example here, but there are hypotheses that these, too, have been built on top of existing road networks in the conquered areas in Western Europe, and that puts their dates of origin a while back still.

There are ancient Roman aqueducts still delivering water to some desert tribes/villages.