Some monitors can downscale if the input resolution exceeds the screen resolution. The result is usually not very pleasant though. Even in the best cases, it tends to look blurry (blurrier than 1600x900) and text difficult to read. Your question implies you've tried it and it doesn't work? In that case there's not much you can do other than get a different monitor.
Back in the Windows XP days, Microsoft let you manually set the desktop resolution independent of screen resolution. So you could set the desktop to 1920x1080, but display it on a 1600x900 monitor at native resolution. The monitor would display only a 1600x900 subsection of the 1920x1080 screen. When you moved the mouse to the edges or corners, the portion of the desktop being displayed would slide over. But unfortunately they got rid of the feature in later versions of Windows. A pity since I rather liked it (made 1366x768 screens tolerable).
If you've got a second PC, you could simulate it by having the second PC output to the monitor at 1600x900. Then have the second PC display the primary PC's screen using VNC (kinda like remote desktop except it doesn't log you out). But even with Gigabit Ethernet the franerate will not be that great. Something like Splashtop (which compresses the desktop into a h.264 video stream) may work better at preserving framerate.