There are art references that cause many to think Paul fell from his horse. One of the most famous of these is Caravaggio’s famous painting titled “Conversion on the Way to Damascus”.(1) Wikipedia (2) suggests the Golden Legend (3) may have framed the event for Caravaggio.
In doing some online research on this question, I discovered some interesting information related to this question.
Dr. Taylor Marshall points out "Pharisees prayed regularly throughout the day in obedience to Psalm 55:16-17, “But I call upon God, and the LORD will save me. Evening and morning and at noon.”"(4)
Mr. Hector Molina, refers to the significance of this happening in daylight, "It was midday ... Some commentators point out that this was to show that there was no delusion from nightly appearances." He also refers to Dr. Marshall's statement about midday being a prayer time for the Pharisees. (5)
There is no mention of Paul riding any animal to Damascus, but this fact in no way can be used to assert he was not. That would be an argument from silence which is an error in reason. Further, it cannot be successfully argued that because it is not explicitly stated it did not happen. One example should suffice: Scripture rarely refers to Joseph, Mary & Jesus eating, but that does not mean they fasted their entire lives. The Catholic Church does not accept the belief that unless something is explicitly stated in Scripture it cannot have happened. The Church accepts and teaches the four senses of Scripture (CCC 115-119), among other information such as "In order to reveal himself to men, in the condescension of his goodness God speaks to them in human words" (CCC 101).
Since Damascus was about 135 miles from Jerusalem, it seems reasonable that Paul was riding on an animal, perhaps a horse or a camel, particularly since he was sent on the very important mission of weeding out all those in "the way". He could have also been walking with his companions, it just seems like a long journey on foot when he was so impassioned about this mission. The prayer time referenced by Dr. Marshall in no way would indicate Paul couldn't have been riding an animal and dismounted to pray. I would expect this of the "Pharisee of Pharisees" as he referred to himself (Phil 3:5).
(5) http://www.catholic.com/blog/hector-molina/did-saul-actually-fall-off-his-horse-on-the-road-to-damascus, accessed 5/1/2015.