I have a mild class 3 malocclusion. How long will it take braces to fix this?

Ask your doctor. A malocclusion varies from one patient to another. Only the doctor treating your case is capable of offering a reasonable estimation of treatment time.
Average 18-24 months. But "mild" is an elusive term. Is your problem dental, skeletal, or a combination of both. Do you have a deep bite or an open bite. Do you have enough room for all your teeth when properly aligned. Is your upper jaw proper width to match lower jaw. All these factors, and so many more, have to be taken into account when estimating treatment time. See an orthodontic specialist.
Depends. It all depends upon the nature of the underbite and the severity. As adults, true, severe skeletal discrepancies are usually dealt with by either extractions of teeth to hide or camouflage and/or orthognathic surgery. The only way to determine the severity of your situation is with an appointment with your orthodontic specialist.
As your orthodontist. Treatment times range widely and there is no one better to answer your question than the doctor who will be treating you.
Need more info. Depends on many factors-- are there crossbites? Rotated teeth? I would consult with 2 or 3 orthodontists who can see your teeth (assuming you got this diagnosis from one orthodontist and are concerned about the time estimate they provided).
How mild? As you are not growing anymore, we can only work with what you have. An internet diagnosis of the severity of your class iii malocclusion is impossible, but there are many ways to approach it conventionally: we now have bone anchors, multi-loop edgewise technique (meaw) and if not too severe you are very likely to be helped. Time? Can't see you from here. Best of luck!