I was answering the question of advantages of bicycles over horses, not on the feasibility of their production in ancient times.
That said, history is interesting.
Adam Smith's Wealth of Nations describes mass-production techniques, including specialization, special tools adapted to single sub-tasks, and a style of production approaching an assembly-line (work produced at stations, with the pieces moving from station to station. In 1776.
The Chinese were employing mass production techniques in the manufacture of crossbos in the Warring States period (475 BC - 201 BC): http://www.ingentaconnect.com/content/maney/aaa/2008/0000000...
Modern mass production dates to 1803 and production of sailing blocks (used to raise and lower rigging on ships) in England, from which it spread to the US for clock and armaments manufacture. Eli Whitney (of cotton gin fame) attempted mass armaments production in 1798 but failed to achieve quantity output for another 8 years. Refinements in metalurgy, scientific management, steam and electric power, etc., produced further advances -- as with most developments, the seeds were planted early and incrementally developed over time.
As I note in another post, the two specific advances that really made the bicycle possible were advanced metalurgy and vulcanized pneumatic tyres. Both of these weren't particularly feasible until they actually emerged. That said, high-quality Damascus steel dates to 300 BC. The Greeks and Romans had and used coal (used in the production of both steel and synthetic rubber). Rubber was known as far back as 1600 BC, but among the Mayans in South America (it wasn't introduced to Europe until 1736): http://www.essortment.com/history-rubber-21100.html
But the biggest trigger for widespread bicycle adoption was probably socioeconomic, not technological. By the late 19th Century, a middle class with both discretionary income and free time had emerged. And though bikes were used for utility (commuting, errends, light transport), the real driver was recreation. The presence of intense socioeconomic stratification, slavery, feudal systems, and the like, had a serious damping effect on both creation and adoption of new technolgies and products.