What's the English equivalent of the Japanese saying, “A fart ruins 100 days of sermons by the priest (bishop)”?

Although neither of these are common English idioms, I'm reminded of Benjamin Franklin's proverb:

It takes many good deeds to build a good reputation, and only one bad one to lose it.

or Warren Buffet's paraphrase:

It takes 20 years to build a reputation and five minutes to ruin it.

Both of these capture the laborious nature of the “100 sermons” in the Japanese original.

Maybe "One rotten apple spoils the barrel" is similar.

For want of a nail is a proverbial rhyme about how a seemingly trivial oversight can have far-reaching consequences.

For want of a nail the shoe was lost.

For want of a shoe the horse was lost.

For want of a horse the rider was lost.

For want of a rider the message was lost.

For want of a message the battle was lost.

For want of a battle the kingdom was lost.

And all for the want of a horseshoe nail.

One idiomatic expression is fly in the ointment.

For five centuries, 'a fly in the ointment' has meant a small defect that spoils something valuable or is a source of annoyance. The modern version [...] suggests that something unpleasant may come or has come to light in a proposition or condition that is almost too pleasing; that there is something wrong hidden, unexpected somewhere.

So as you see, the original meaning is pretty much the one you're looking for, but the current meaning is rather different, putting additional stress on the defect not having been apparent right from the start.

As the Wikipedia article mentions, the source is likely Ecclesiastes 10:1. In the King James version, the passage reads, "Dead flies cause the ointment of the apothecary to send forth a stinking savour: so doth a little folly him that is in reputation for wisdom and honour."

In Russian, there's the expression "a spoon of tar ruins a barrel of honey", but I don't think I ever saw anything similar in English; I'd certainly remember it.