If you are living in a two-floor apartment, then odds are that you've got a few nooks and crannies on that second floor, near the staircase. While it's not uncommon to find everyday uses for these odd spaces, we're really fans of turning these into small, but capable, workstations.
Placing an office high up has the added benefit of giving you a great view while you're working. For some people, it can be distracting, but for others it's a nice add-on to the home office setup. We've lived in such an apartment and usually, these areas are rarely used well. The best way would be to place a desk on or near the railing that faces the window. This can be counterintuitive, as these spaces are usually just glorified hallways for other rooms or the patio.
Chuck Rifici had such a nook that he turned into a home office. His mini second floor was just the right size for his computer workstation. While his office feels like a private space, since there's nothing else up there, for most lofts and two-story apartments this won't be the case. The trick is to place some furniture around to cordon off the office. Wall dividers can also be used, but in all honesty, they won't look that appealing in this setting since they'll most probably cut the natural light off. A filing cabinet, a smaller desk, a low bookcase, even a sofa would do well.
Chuck uses three Ergotron LX monitor arms, keeping the computer monitors off the desk. This allows him to store items beneath them, like peripherals. Another great idea would be to integrate some storage into and under the desk, using movable drawers and racks. This would allow you to put all of your peripherals out of sight, which is always nice if your space isn't that private. The ideas that work well in a closet home office also work well here.