Well, it's great that you're learning with Vonnegut but you have to go in knowing that he's going to be playful with language and frequently ironic.
His ulcer gave him a twinge...
This should be straightforward.
...the ulcer that had grown in size and authority over the years...
Here, the authority is an odd word that catches the reader and personifies the ulcer. It gives it the aura of a parent, a teacher, a policeman, a pastor, or governor, which is odd. At the same time, it's understandable: as the ulcer becomes stronger, it exerts more control over the man and his life.
The man's authority didn't necessarily grow: the ulcer's power over his life did.
...over the years of his career as an interpreter of America to provincial and ignorant notables from backwaters of civilization.
This section has nothing to do with the authority, except by ironic contrast.
...of his career... modifies the years and, given the timing, indirectly states that his ulcer is due to his career.
...as an interpreter... modifies the career and explains what it is.
You seem to be taking it literally, though, and ...an interpreter of America... is obviously not a translator of languages but a tour guide, cultural professor, &c. who metaphorically acts as an interpreter.
...to provincial and ignorant notables from backwaters of civilization explains to whom he is interpreting America. On the surface, it's a very jingoistic way to phrase it that may make the author seem to be very pro-American. In fact, given (a) that this is Vonnegut, (b) the content of the rest of this book, and (c) the content of the rest of this sentence (the man's job is so unhealthy as to have taken control of his life via his ulcer), it is unquestionably ironic and anti-American. He is expressing the views of the man, but in a way that undercuts it and makes the reader question how such a destructive lifestyle could really be superior to others.