use a roller, they're about $1 for a walmart cover - which I find to
be adequate, I don't care what anyone says, I"m not going to spend $8
for a roller cover when I can use a $1 one that looks great.

I've painted over texture before, it takes a very heavy nap roller
cover. These are about $1.50 a piece at walmart, they come in three
packs. Just load that roller up with paint and roll with a little
extra pressure...it's not much different than painting a normal
wall. You may have to go back and touch a few areas with a brush,
you'll see what I mean after you start.

Don't ever use a sprayer. If you're a professional painter and are
painting an entire house one color and it has no floor covering or
trim, that's the only way I could see it being a time savings.

Here's a little tip for you though, if you're really into saving
time... have a helper. For some reason, two guys can get the job
done in a fourth of the time, from masking the flooring to clean up.
Also, if the ceiling is at all questionable, do it at the same time as
the walls because you're masking the floor anyway and you'll paint
over any splatter onto the walls right away so you don't have to worry
about them.

Good luck, If you really focus, the job won't take near as long as you
think it will.