Given answer in book is (E) "No Error". But there is a rule in textbook that we can't use article before the noun which comes after "Each / Every / Each and Every / Either / Neither". Can anybody help in understanding the answer?
There is no error. What is in error is the rule in your text book or your interpretation of the rule.
In the passage provided, the meaning is
Each of the ten capitains had an achievment of his own.
Notice that "each" refers to "captains," and "captains" has a definite article associated with it. For emphasis, you may also say
The ten captains each had an achievement of his own
The ten captains had each an achievement of his own
although the second is very rare.
What is not rare is
Each captain had an achievement of his own.
What all these examples share is that "each" refers to individual members of a group of capitains, not their individual achievements.
What you cannot grammatically say is
Each the ten captains had ...
The rule is that if "each" precedes the noun to which "each" applies, it cannot be immediately followed by an article applying to the same noun. Moreover, that rule does not apply to "either" or "neither.
Either the man or the boy can do it because each knows the password
Neither the man nor the boy can do it because neither one knows the password.