As others have noted, there are multiple cultures where the most "honored" warriors were expected to be highly proficient with bows. Romans, Persians, Japanese, Chinese, Mongols and many others. Henry VIII was even an experienced archer, in addition to the laws he (and his predecessors) passed to encourage archery in their country. Even the French have some strong archery traditions, but they were greatly overshadowed by the English.
All this tells me that the bow has never been considered a "dishonorable" weapon. On the other hand, when two opponents of stature meet, the "proper" way to fight is in melee. A knight, or knightly paladin, who issues a challenge to an opponent (on a battlefield or off) would probably do with a melee weapon.
Besides the horse, the bow was one of the most prevalent and powerful weapons available throughout most of history. Therefore, it seems silly that a martial caste of society would ignore it.
Well said Gru. I find that there are many here being "subjective" to the differing cultures of Paladins. Pathfinder leaves it quite open to create a Paladin from differing cultures and not limited to an image of Christian Crusaders. Was not Genghis Khan revered by the Mongols as their holy warrior empowered by their god? What of the Persian Immortals, who also wielded bows in service to their divine god. I also find it interesting that the point of shooting down an animal or monster with a bow is fine, but somehow many forget to watch the player behind the bow and his/her reactions. If my player shot down a defenseless villain willing to give up after they asked for mercy, i'd say highly dishonorable. They should be brought to justice. I'd also tell an NPC to "Drop their sword" so they could be brought to justice. If they lunge at me, then face my Paladin gods light. lol