Assuming it is clear for which reason a president of the United States should be impeached...
If it is very clear, and there is no significant controversy, then it could be done in a matter of days, perhaps a week or two at most. None of the steps in the process has a minimum duration.
Imagine a president who clearly offered to sign or veto a bill in exchange for a large sum of money, and that the evidence is clear and not subject to dispute. That is bribery, one of the two offences that are explicit grounds for impeachment.
The house judiciary committee could write up an article of impeachment based on the evidence and submit it to the house, which could pass it and send it to the senate, probably all in the same day.
The senate might need a day or so to organize the trial, but they might not. The evidence, being clear, should not take very long to present in the trial. The most uncertain element would be the time taken by the president to present a defense.
According to Wikipedia, the Clinton trial spent two days in procedural matters on January 7th and 8th. The trial remained in recess while awaiting briefs; Clinton's brief was submitted on Wednesday the 13th. The defense presented its case over three days, from the 19th to the 21st.
The Clinton case was more controversial and complicated than the hypothetical case assumed here, so we can say that the defense would probably consume less than the 8 days in Clinton's case. The rest of the time in the Clinton case was consumed by questions from the senators to the parties in the case, debate about whether to dismiss the charges and whether to call witnesses, videotaping testimony of witnesses, presenting excerpts of the testimony to the senators, and hearing closing arguments. These things could also take far less time, in some cases none at all, because of the lack of controversy.
This puts an upper limit on the duration of an "open-and-shut" case at a couple of weeks, but it could probably be done in a week or even less, especially if the Senate adopted procedural rules to limit dilatory maneuvers by the president.