When you're unsure which article to use, or whether to use an article at all, it's good to start with looking at the noun: is it countable, or is it uncountable? Is it singular or plural?
After a few minutes of (a) futile argument we had to accept...
The noun "argument" is a noncount noun here, and means "the process of discussing something, arguing over something".
Hence, it's better not to use "a" there, because the indefinite article is used with singular count nouns.
After several hours of fighting the enemy withdrew.
Your "argument" is close in meaning to "fighting".
The word "argument" has countable senses too: "a fact or statement used to support a proposition; a reason":
It is hard to imagine the court would seriously consider such an argument against otherwise-permissible regulation of guns.
He has the reputation of (a) troublemaker.
"Troublemaker" is a singular count noun, and we can use "a" here. We cannot just omit the use of articles here.
We cannot use "the", because we have in mind a "prototypical troublemaker".
Also note that you should use "the" before "reputation", because otherwise it might mean that the prototypical troublemaker has several reputations, and your subject has one of the several possible reputations.