The first sentence "I am neither robber nor police." is the correct form, but the way that I would say it is" "I am neither a robber nor the police."
The problem with the second sentence is that the verb is out of place. A Neither/Nor phrases represents two possible ways to end a sentence:
"I am not a robber." and "I am not the police.", but then combines them into the same sentence.
Using an Either/Or or Neither/Nor phrase is a way of combining two different sentences into one. This sentence combines "I am not a robber." with "I am not the police.". The combined sentence now has one beginning: "I am not..." and two endings: "...a robber." and "...the police.".
Either/Or lets you have two endings to the same sentence, and so it becomes: "I am not either a robber or the police." This sounds awkward to our ears which is why we have Neither/Nor for negatives. The "not" distributes over the either and the or and we get neither and nor and thus:
"I am neither a robber nor the police."