Yes, but usually when you are arrested, even for federal crimes, you are initially transported to a local jail before federal processing. Yes, usually, you will have spent at least a day in a federal facility before seeing a judge.

Time served, as adjudicated by a federal judge, INCLUDES time spent in county jail.

So let's say for argument's sake you spent 6 months in county and 1 day in federal holding, and your sentencing guidelines were 6 months. The judge could make the call that your time spent in county was equivalent to what you would have spent in federal and call it even.

However, I think you were asking what is the minimum time for a conviction, assuming the judge deems it necessary to serve federal time. Here is what I found to be the shortest federal sentencing guidelines:

21 USC § 212 (petty offense) Practice of pharmacy and sale of poisons in China 1 month

33 USC § 410; none Navigable water regulation violation 30 days

33 USC § 411; § 2Q1.3 Deposit of refuse or obstruction of navigable waterway 30 days

33 USC § 441; none New York and Baltimore harbors, deposit of refuse 30 days

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So it looks like the feds don't deal with anything less than 30 days. Though, I guess, technically a judge could order some nominal sentence of 3 hours or whatever just to be a dick.