Short answer: A centaur can't reach his own hooves, but can reach the hooves of others by lying down or kneeling. One centaur can maintain the hooves of another.
Better answer: There are a few reasons why horses wear shoes: to keep their hooves from cracking on paved roads; to gain better grip on cobblestones, in slippery conditions, or during races; and fashion (some horse shoes have a leather portion against the hoof to make them look nice for horse shows.) Centaurs would wear shoes for the exact same reasons.
If centaurs built a city, I doubt they would pave the roads, per se. They would probably install something akin to a polo pitch, which could be tended by horses, goats, etc. who would eat the grass and keep the pitch clear for centaurs to gallop through.
Hale and hearty centaurs would likely not need shoes or a farrier because they would take plenty of exercise in the fields where they don't need shoes and will wear down the hooves enough that they don't need trimming. Sick centaurs who can't run around for a time would grow hooves that would need trimming. The need for a farrier would probably be associated with poor health and would thus be stigmatized.
For humans, long fingernails are associated with a life of leisure because one cannot work manual labor and maintain long nails. You might think this would be the same for centaurs and their hooves but it only applies to their fingernails. The hooves are a centaur's toenails and nobody has anything good to say about long toenails. In fact, excessively grown hooves can cripple a horse or centaur.
Because shoes can only be installed by another centaur, they would be essential indicators of civility and sociability - you can't have shoes if you can't get someone willing to put them on you. Shoes would be a status symbol. The fancier and more difficult they are to put on, the better, because it means that someone else took a lot of time to service your feet.
Barefoot centaurs would include: barbarians/wild centaurs; low class, poor, or rural centaurs; those who prefer to live a simpler life (hippy centaurs); and higher class centaurs who are counter-cultural for some reason (maybe in protest).
Rich, high status centaurs would flaunt their wealth and status with elaborate, difficult shoes. Military centaurs would have studded shoes for difficult terrain. Athletes would have specialize shoes for their sport. (Race horses wear lightweight shoes for racing. These wear out after just one use. Their only purpose is to reinforce the foot for the duration of the race.) Injured centaurs would wear remedial shoes to repair their gait and help them heal.