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This question is hard to answer since you would have to define "in use" very precisely. Here are some wild guesses:

• Wikipedia has a list of the oldest buildings in the world. Prominently feature tombs/graves/similar and you could well argue they are still in use. That would go back as far as very roughly 4000 B.C.


• 
  Stonehenge is believed to have been constructed ca. 3000 - 2000 B.C. and still a place of religious significance for e.g. neopaganism.


• Parts of the Great Stupa of Sanchi are from as early as 300 B.C. and the complex is still in use as a temple.


• The Verona Arena is dated 30 A.D. and still used e.g. for concerts.

Further reading:

• 15 of the World's Oldest Buildings Still in Use Today


• The same question on History SE: Oldest building in the world still in use
  

The accepted answer there is the Pantheon in Rome from 128 A.D. and still a church to day. However the most upvoted answer points to several arenas/theaters still in occasional use.

The Epidaurus Theatre (ca. 300-340 BC), the Delphi theatre (4th century BC) and the Odeon of Herodes Atticus (161 AD) in the Acropolis of Athens (known locally as the the Herodeon), still fulfil their original purpose, all three are constantly used as venues for various festivals. The ancient theatre in Dion is also used occasionally.

The Theatre of Epidaurus (4th century BCE!) still has performances. It is not, however, an enclosed building.

The Great Pyramid of Giza initially had two functions, to serve as a tomb for the pharaoh, and to demonstrate his power and prestige. Today, it has two functions, to serve as a tourist attraction and to demonstrate the power and prestige of the pharaohs. It is a structure that has continuously fulfilled one of its functions since 2540 BC.

As far as I know, it is never closed to tourists to fulfill its status symbol function, but it makes quite a good status symbol even while open.

The Etchmiadzin Cathedral was built in 301 AD.

The original church was built in the early fourth century—between 301 and 303 according to tradition—by Armenia's patron saint Gregory the Illuminator, following the adoption of Christianity as a state religion by King Tiridates III. It replaced a preexisting temple, symbolizing the conversion from paganism to Christianity. The core of the current building was built in 483/4 by Vahan Mamikonian after the cathedral was severely damaged in a Persian invasion. From its foundation until the second half of the fifth century, Etchmiadzin was the seat of the Catholicos, the supreme head of the Armenian Church.

I have prayed at the Western Wall of The Temple Mount in Jerusalem. My ancestors have been priests there since about the 10th century BC.

A lot depends on your definitions of "building", "use", and whether "use" is required to be continuous throughout history. A good candidate is the pantheon in Rome (AD 128). It was a place of pagan worship to start with, became a church when Rome became Christian, and has been in use as a church since then. Amphitheatres and pyramids don't really fit the "walls supporting a roof" definition of "building".

The cathedral in Syracuse was built as a temple to Athena? around 500BCE, and converted into a church around 500CE