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Camels are unclean animals. A Jew in that era would be about as likely to eat pork as to ride a camel. Many probably did, but it seems unlikely.
Leviticus 11:4 The camel, because it chews its cud but does not part its hoof, is unclean to you.
They traveled by horse (or Paul, especially, would have been apoplectic). There is a cleansing ritual, also chapter 11 of Leviticus, so in a matter of expediency there may have been an exception. Still, it seems far more likely they traveled by horse.
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This is actually an interesting topic because there is much debate about it!
Tradition has held that Saul/Paul rode a horse to Damascus, but when you read the verses that deal with this, a horse is not mentioned. So there seems to be a hint in tradition that he rode a horse but scripture does not back it up. So here it depends on how much you rely on tradition and the origins of this specific one.
Similarly, our nativity scenes (tradition) depict the wise men as riding camels. They were foreigners, not Jews, but this knowledge mixed with a knowledge of the middle east as being inhabited by horses and camels suggests they could have ridden either. It was also common to ride donkeys or asses (see John chapter 12, and this post).