Buddhism is older than Christianity. The historical Buddha is generally placed in the first millennium BCE, several centuries before Jesus of Nazareth and the emergence of Christianity in the first century CE. Exact dates for the Buddha remain debated, but the chronological order is clear.

When did Buddhism begin?
Buddhism developed from the teachings of Siddhartha Gautama, the Buddha, in ancient northern India. Traditional chronologies often place his life around the sixth to fifth centuries BCE, while modern scholarship debates the exact dates and in many cases places his death later than older traditional calculations.
After his death, communities of monks, nuns, and lay followers preserved teachings orally and developed monastic institutions. Buddhism spread across South Asia and later into Sri Lanka, Central Asia, Southeast Asia, China, Korea, Japan, Tibet, and other regions.
When did Christianity begin?
Christianity arose in the first century CE from the life, teachings, death, and proclaimed resurrection of Jesus of Nazareth. Jesus was a Jew living in Roman Judea, and the earliest Christian movement began within a Jewish setting.
His followers proclaimed him as Messiah and Lord and formed communities throughout the eastern Mediterranean. The writings that became the New Testament were produced during the first century and early second century.
How much older is Buddhism?
If the Buddha is placed broadly in the fifth century BCE and Jesus in the early first century CE, Buddhism predates Christianity by roughly four to five centuries. The exact number depends on which scholarly chronology of the Buddha is used.
It is better to give a range than to claim an exact gap based on uncertain ancient dates.
Older does not mean more true
Chronological age does not establish the truth or superiority of a religion. Religious traditions make different claims about revelation, awakening, God, salvation, human nature, and the goal of life. Historical sequence can be studied without ranking spiritual value.
Likewise, later religions can draw on older traditions, reinterpret them, or arise independently.
Did Buddhism influence early Christianity?
Comparisons between Jesus and the Buddha have produced many modern books and theories. Direct historical influence on the formation of early Christianity has not been established by evidence strong enough to be treated as consensus.
There were ancient trade routes linking the Mediterranean and Asia, so contact between cultures was possible. Possibility, however, is not the same as proof of direct doctrinal borrowing.
Two different historical contexts
Buddhism arose amid South Asian religious debates about karma, rebirth, asceticism, ritual, and liberation. Christianity arose within Second Temple Judaism under Roman rule and developed around Jewish Scripture, messianic expectation, and claims about Jesus.
Understanding those different contexts is more informative than simply asking which religion is older.
See Buddha, Jesus Christ, Buddhism, and Christianity.