Typically, I just rinse off the brass with tap water. If your water is relatively soft, you'll have no problems. After rinsing, I just dump the cases into a towel, and use the "bowling ball" method of removing the excess water. Then into an oven on the "warm" setting for about 20 mins.
BTW: The "bowling ball" method is simply dropping the cases onto a large towel, then folding each side (lengthwise) over the brass. Holding each end of the towel (to keep the flaps folded over the brass, you then roll the brass from one end of the towel to the other, keeping the brass in the (now made) tube of a towel). This wipes all the excess water off the exterior of the case, as well as knocks water free from inside the case. After rolling the cases back and forth several times, just move one hand over an old cookie sheet, and let the cases slide out of the end of the towel onto the cookie sheet. Make sure they're laying flat, and toss into the oven. This will also dislodge any remaining pins that may have adhered to the inside of the case due to any water inside the case causing surface tension to stick pins to the inside case wall.
And yes, that looks like the right kind of car wash (looks very much like the "Meguir's brand we have here).