他的签证很快就办理好了。

If you must have the three words together, then, it more or less like "他需要办签证, 很快就办下来了". Roughly means, he needs to apply for a visa. It is approved soon after his application. We normally would say something either as Walter mentioned above or like "他的签证很快就出来了", "使馆很快就处理了他的签证申请." By directly translated from English, your sentence is “他的签证被处理的很快". It sounds rather odd in Chinese perspective, as the passive voice is not as common in Mandarin as it is in English.

A native way to convey the same meaning: 他的签证很快就办下来了