Just did this exact job and used an oil based stain and a roller--small one that fits the with of one slat. Easier than a brush and good coverage. Never a fun job, used a brush for cutting in, but the bulk was rolled.

Paint the back side? If you don't want cupping of the slats, you should seal the back also, especially if either side is subject to much direct sun. Did I do that? Hell no--too much extra work, but I knew I should have!

If you have the chance (sometimes its covered up where you have no access) you should always seal both side. If one side of the board is seal, the piece will suck up moisture in the air at different rate causing it to cup or warp.