Thanks for the feedback and questions!
Yep, this is the DIY version of the Road Shower Cole sells--which is a pretty sweet product, for sure. Changing the shape of the tank allows the same volume in less linear feet, which is nice. Honestly, if I had the extra cash I could justify for it, I would seriously consider getting one of those. The new version that is coming out is nicer than the original, which was introduced about 5-7 years ago, I think?
As to how to pressurize--here's a pic of the cap with a bicycle Presta valve fitted to it. The idea is to be able to pressurize it with a hand-held bike pump, or if you use a Schrader valve (or an adapter) and have an air compressor, you could pressurize it that way. You don't need much pressure--maybe 15 lbs?
Also, you don't *need* to pressurize it--you could definitely just use it gravity fed, especially with it being on top of a van! The pressure is nice for times you may want to use it for other things, like cleaning off a bike or spraying down something where pressure would help. If the tank was mounted lower, say on a car roof lower than head height, than pressurizing would be handy unless you want to crouch down to rinse your hair.