The halachos of washing for bread are more complicated. I believe there may be a problem of using a water bottle or paper cup (although I’m not sure). For waking up from sleeping, a water bottle or disposable cup are fine.
“Do you have to wash if you didn’t sleep in a bed?”
I don’t know for sure, but I think if it’s “sheina kavua” you probably do. But I just realized that she did say “nap”, so I don’t know. I was told that it’s not clear that I have to wash if I sleep on a bus, but I don’t think I was told that I definitely don’t.
I was concerned that it might be a problem to wash outside because you’re not supposed to wash on the ground, and I was told that since it wasn’t clear that I have to wash at all, I didn’t have to worry about it (taking into account that I try to find a place at the edge of the sidewalk where people aren’t likely to be walking).
In any case if it’s nighttime, and it’s the only sleep you’re getting even if you are calling it a nap, I wonder if it would be different.
In any case, you would have to wash negelvasser at alos hashachar regardless of where and how much you slept.
Nesiah Tova!