Some Preliminary Words...
I'm not entirely sure that withholding a character's name is the best way to go about producing an aura of mystery. In fact, this is something that I've seen in a lot of early writers (and I even did it a few times myself back in high school), but which almost never works. Usually, the reader just finds it incredibly annoying. Even if you can pull off the "no-name mystery" grammatically so that it doesn't sound awkward or repetitive, the reader would almost always like something to remember him by.
A lot of writers (I struggle with this, too) decide to retain little details - name, location, whatever - to try for an air of mystery. While we have all probably read books that do this at the beginning, it's usually not for very long - rarely for entire stories. If the aura of mystery hinges on these details, then it's not going to be mysterious enough to keep the reader reading. It will just be annoying.
My Answer, Though...
If I were you, I would work on making sure your story has such a strong "mystery" element to it that you could reveal more about your character if you wanted to, without destroying that element. Then, you can decide what you want to reveal and what you want to withhold, and perhaps the mystery elements themselves will lend you ideas.
In this case, you wouldn't need to give his real name. He could have an alias, too, just like his boss. Just one idea.
Another is to consider the first person perspective. Difficult, but if he is always thinking of himself in terms of "I" and "me" and doesn't run into anybody who uses his name, then that could work.
Regardless of whether you add an alias or write from first person, I would (if I were you) make sure that your information you're giving has more of his attitude behind it. If we aren't allowed to know your character, we should at least "experience" him. Rather than simply being told about his job, we should almost definitely find out about his work via his attitude about it - or something along those liens. If you want to maintain that aura of mystery, then we'll have to be invested in the character enough to care about him, even if we don't know his name.
Definitely eliminate any unnecessary words - include "he" and "him".
In other words, without a name, you gotta pull double duty to keep the reader sucked in! That alone could potentially overcome the repetitiveness of "he" and "him" and such. If done well enough. This won't be great (and it definitely has my tone and attitude thrown in), but here's a quick example of what I mean:
The alarm beeped.
His eyes crept open as he slowly roused himself from his light slumber; he had work to do, and it was time to begin.
The damn job was too stressful, with too little rewards, but it wasn't like he had a choice. Just a little pawn in a large game of chess. A pawn who doesn't even know his king. As one of the few members of such a small organization, most people would have assumed he knew his boss. He'd never even met the man. No, he was expected to do the work of ten men, communicate solely by e-mail and text, and chug along as best he could with the orders he was given.
You know your story, though, so I'm sure you can think of something much more imaginative (and mysterious!) than that. :-)