Or do I?

[or do I?] is not an idiom, but a fairly popular writing style, which involves making what appears to be an afirmative statement, followed by a question against that statement, to create the same effect of stating uncertainty or indecision on that statement.
In your example, the prose says: [It's a dog-eat-dog world and I have to stay on top of it- or do I?]. This is equivalent to saying: Although it's a dog-eat-dog world, I really don't know whether I necessarily have to stay on top of it. The second sentence [I don't seem to have answers to anything any more] further emphasizes the author's uncertainty about what role to play in this dog-eat-dog world.
[or do I?] can take other similar forms, such as [or should I?], [or can he?] etc. For example: We can always forgive - or can we? This is equal to saying [can we always forgive?]

(It's a dog-eat-dog world) is an idiom but I have never seen the rest of the sentense above. It seems to me (or do I?) refers to (or do I have to?) My guess is that it might mean (do I really have to stay on the top of it?). You can feel the confusion of the writer from the second sentense.