This is the issue with trying to build some equipment like this by yourself.
You have to source things from multiple vendors, which means shipping charges.
You buy a pump someplace, and then have to pay to ship it, you find the nozzles somewhere else, and then have to pay to ship them, then you use gas driving all over town trying to find the rest of the plumbing, find it and then you have to pay tax on it, drive home, put it together.. you find out you forgot something, so more gas and time.
You get it all together, add it up, and you probably spent just as much if not more than you would have buying a herpmist, plus a lot more time.
The guys who sell these online buy in bulk, get discounts, save on shipping, etc.
If you want to use an air pump to power your misting system, it would go something like this.
1) The air pump would pump air into a sealed pressure vessel filled partway with water, bringing it up to a high psi, assuming your air pump was capable of this.
2) You would need some kind of check valve to prevent it from over pressurizing, and some kind of controller that would turn the air pump off at a certain psi.
3) some kind of solenoid valve would have to be attached to the plumbing and controlled by a lawn timer
4) from there it would have to go to misting nozzles, etc.