Assuming they observed its birthday or otherwise counted Its years.
The founding date of the city of Rome (ab urbe condita) was not always agreed upon, but by the time of Philip the Arab, it was commonly held to be April 21, 753 BC.
In 248 AD, there were massive festivities, including three days and three nights of the Secular Games (Ludi Saeculares / Ludi Terentini). They were celebrations held irregularly with connections to 100 or 110 year divisions (Saeculum). Various leaders changed the dates to suit their needs, postponing, ignoring, or bringing forward the dates, so that the actual dates of the games were BC 17, AD 47, 88, 148, 204, and 248. And some others that were canceled due to war, strife, etc.
To get a sense of scale, Wikipedia says that in the Colosseum during the AD 248 games:
more than 1,000 gladiators were killed along with hundreds of exotic animals including hippos, leopards, lions, giraffes, and one rhinoceros.
But I can’t verify that figure, it’s only sourced in a TV show. Hopefully somebody has another source.
Similarly:
During the secular games of AD 204, Septimius Severus arranged a venatio, probably in the Circus Maximus, which consisted of a ship which fell apart to release bears, lions and lionesses, leopards, ostriches, wild asses, and bison.
[Wiedemann. Emperors and Gladiators. 1995]
What festivities were at the AD 248 games, and in which locations? I don’t know exactly, and the Secular Games held by Augustus in 17 BC were probably more elaborate than the ones by Philip the Arab. But you could expect most of these events throughout the city and empire regardless of which Secular Games you were alive for:
[Robinson. Ancient Rome - City Planning and Administration. 1994]
Besides these displays, there were altars erected or unburied, animals sacrificed, songs and prayers directed at the gods, grains and food distributed to the entertainers. Many tens (or hundreds) of thousands of people must have flocked to Rome for the celebrations. I wonder if the city smelled more of festival food, death, or sewage.
As a follow up, did the Byzantines still count from the founding of Rome and celebrate their 2nd millennium in the 1200s? Or celebrate 1000 in some way?