It depends on the type of wood and existing treatment on it. If it already has paint on it and you are painting a similar color it won't need as many coats as fresh wood.
You have about 3100 SF to paint and you don't have any edging to worry about and it sounds like the everything is accessible. When you are painting outside you can be a little messier than inside a house so its not unreasonable to get 400-600SF painted in an hour when hand painting. Which would get you about one coat a day on the fence.
Paint sprayers are great if you are doing a big area, but outside can be a burden with wind, bugs and pollen. They also typically put the paint on thinner than if you roll or brush it on. Or you can end up spraying to much on and needing to smooth it out with a roller.
Depending on what sprayer you get they can also require a fair amount of clean up and setup which is something to consider. The first time I used a sprayer I covered myself in paint, made a mess, and generally created way more work for myself than needed. After that I was way more cautious about completely prepping areas before hand, getting the right spray consistency and not rushing myself with the sprayer.
I've found the sprayer is best for interior work laying down primer or white base coats before flooring or a lot of other finishes are in.