What does 'saw myself going' mean here?
I saw on the door surgery began at 8:30 and I ought to have guessed there would be a lot of people as usual, but for some reason I just saw myself going in and seeing the doctor straight off.
3 a : to form a mental picture of : visualize
So you can parse this as
I visualized (what did I visualize?)
myself (doing what?)
In this sort of context what you're reading is commentary on a prediction, not literal experiences.
They constructed an erroneous scenario in their mind where they would be able to go straight into the doctor's office rather than, for example, waiting an hour.
This usage of See is almost always in a predictive scenario-driven context, similar to Oracles "seeing" the future.