I hold that every man should own a compressor, the more powerful, the better.This is a lifetime investment as are all the necessary tools..
Much debate, of course,on how much power, and the source..
How far is the fence from a power source ? What are the future needs ?
I for one am leery of the "bargains" on eBay. Good used compressors are very hard to find.
The oil-less type of compressor may be OK for very light, restricted use.

If the only reason for purchasing the compressor is for the nail gun, why don't you consider a Paslode nail gun. It runs on butane and doesn't need a compressor.

The size of 1" pickets suggests a more decorative fence than a security and containment idea. These pickets are likely to be a bit thinner than 4" or 6" fence pickets.
I would recommend either a narrow-guage (1/4") or standard-guage (1/2") stapler to fasten the pickets to the rails. The rails could be screwed to the posts rather than using a framing nailer to connect them.
Most compressors for pneumatic nailer/stapler use are in the $240 range. I recommend an oil-lubricated compressor rather than an oilless.

Thank you for the feedback, but I don't think I was clear enough. The pickets that I will be using are 1" thick and 6" wide. Would an 18 gauge 2" brad nail be sufficient to nail these pickets?

You need some type of head to keep the "pickets" in place. A staple or a full size nail (with head) is best. The brad and finish nails don't have heads big enough to do the job.

If you do use a small nail like that, one thump on the fence and out pops the "pickets" because the nail pulled through the wood.

If you are going to do the one job only, why not rent ? A brad nailer will NOT suffice for a fencing job and, if the wood you are using is cedar then use galvanized or stainless steel nails. Beware the "oiless compressor" they are strictly a light duty , short life span item.

OK, 6" pickets it is. You will need either a 1-3/4 or 1-7/8 nail so that when you attach the picket to the face of a 2x4, the point of the nail will not protrude on the back side of the 2x4 when the nail is countersunk. Otherwise, you got a heap of grinding to remove all the protruding points.
The coil nailer that you want to rent is called either a siding nailer or a fencing nailer. My recommendation is the Hitachi NV65AH Siding Coil Nailer, bounce-fire trigger.
For the fence framing, get any name-brand framing nailer and load it with 3" galvanized ringshank framing nails. Those posts will vibrate too much if you nail onto them with a hammer.
Get 100' of air hose and the compressor of your choice.