How can I paint vertical surfaces without getting paint runs?
You should be working the paint out to the point that it doesn't run. (People paint walls every day without this problem.) Or buy better paint. (Cheap paint is very watery.) 

You may also be applying too heavily. Put the paint on the surface and brush or roll it outward to distribute. You should have an even coat that's as thin as it can be without showing drag marks, bare spots, etc. If you do that, it won't run.

I like to work from one side to the other, applying paint in a leapfrog manner: Dip the brush, scrape one side clean on the container, dab the surface a few inches away from the previously painted area, and work back to it, blending. With a little experience you can actually feel when you have too much or too little paint on an area by the drag on the brush or the stickiness of the roller.