You know, you can wash a car with a couple buckets of clean water. Makes very little mess and it's fairly fast. Being it's a tiny car I wouldn't think it take much effort. If the car is really dirty, go to a spray wash, drop a couple of quarters and get the big stuff off, but for light washes, the bucket method works great.
Having said that, I'd petition you apartment complex to designate a car washing area. I had two at the one I lived at in Irvine several years ago. They had a hose out to use just for that.