Roman aqueducts essentially never stopped being repaired (I will take aqueduct to also include the system of pipes which fed and distributed water to and from the more recognisable concrete structures). From the moment they were constructed they underwent almost constant checks and maintenance (see, for example, the surviving records of Sextus Frontinus, a water commissioner). There is a vast body of ‘water law’ in Roman sources (you can read some in Codex Iustinianus XI.43) that discuss the entitlement of customers to a given amount of flowing water (as well as the consequences for attempting to steal extra water by widening pipes). The level of maintenance and repair required would have varied widely depending on the particular aqueduct. Some were well-constructed in stable, easy terrain with good water supplies. These could be in use well into the medieval period by successor states which completely lacked the capacity to repair damaged aqueducts (Javier Martínez Jiménez, Reuse, Repair and Reconstruction. Functioning aqueducts in post-Roman Spain, 2012). Others required constant cleaning due to the build-up of sinter, carbonate deposits caused by channelling from sources of hard water.

Rome had a dedicated force of repair people and administrators dedicated to keeping track of maintenance, replacing damaged sections of pipe, checking for fraud (which was constant), stamping pipes with their official diameter, etc. There are also numerous sealed openings in aqueduct channels that suggest they MAY have been used for maintenance access, but archaeologists are yet to identify with any certainty what tools may been used for snaking pipes and water channels. However, aside from this constant, ongoing maintenance, there were also larger scale repairs required at times. Although we don’t have a great abundance of evidence about this, we do have records about the Aqua Claudia, finished in AD 52. It went through substantial repairs in AD 71 and AD 81, being closed for years at a time (Frontinus, as above). The exact reasons for the repairs are unknown but given the length of closure it’s likely that the issue was related to the concrete construction itself, such as cracks and long term leakage. Given that Roman law makes reference to customer entitlement to water paid for, it’s highly likely that large scale repairs requiring aqueduct closure were rare and kept to a minimum.

As for when they stopped functioning, this was again highly variable. Some were deteriorating with the collapse of the Empire while others survived for centuries, though non-functioning in most cases. There are many examples of later empires, as late as the Eighth Century, who made use of the existing, though damaged, aqueducts to service palaces, mills and public fountains (Jiménez, as above). This later usage was mostly private though, I can't think of any examples where they were later put to extensive public use, mostly due to the absolutely enormous public service necessary to maintain them.