Learn Italian: Il dottore arriva fra un attimo. - The doctor is arriving in a moment.

While fra (and its twin word tra) often means "between" when talking about physical space, it has a special meaning when talking about time. When you place fra before a period of time, it means that an event will happen "in" that amount of time in the future. So, fra un attimo means "in a moment".

The English translation says "is arriving", but isn't arriva just the simple present "arrives"?

Yes, arriva is the simple present tense. However, in Italian, the simple present is frequently used to talk about actions happening right now or in the very near future. You don't need a complicated verb form here; using the simple present to mean "is arriving" is perfectly natural and very common.

What exactly is an attimo?

Un attimo literally means an "instant" or a "moment." While it's not a strict measurement of time on a clock, Italians use it constantly to describe something happening very soon, just like saying "in a second" or "in a flash" in English.

How would this sentence change if the doctor was a woman?

You would need to change the masculine noun il dottore to its feminine version, which is la dottoressa. The rest of the sentence would stay exactly the same: La dottoressa arriva fra un attimo.