The first time we see Batman pull off his iconic vanishing act was back in 1972.
In Detective Comics: Issue #424 (1972), Batman is discussing with Commissioner Gordon about a murder of a stockbroker. At this time, Batman vanishes for the first time (image below).
Detective Comics #424 (Page 3-4)
This is the first time he does it in any media (movies, ...
He has extensive training of spiritual and physical martial arts as a peak human and obviously on his many fighting quests he has developed the ability to maneuver his face enough to look different to himself normally like a actor.
Since many depictions of Gotham City show it having a near-constant cloud cover and Batman's preferred working time is the night shift, he wouldn't need much sun screen.
I mean, the Bat Signal, his famous call sign, would only work in a cloudy night.