1. Download "dxwnd" from http://sourceforge.net/projects/dxwnd/ unpack it and start dxwnd.exe

2. Right-click on the white space in dxwnd and select "Add". Make sure "Path:" links to the PrinceofPersia_Launcher.exe; after selecting that .exe the path should look like that C:\Program Files (x86)\Steam\Steamapps\common\Prince of Persia\PrinceofPersia_Launcher.EXE, this obviously differs if you play the GOG version or if you installed the game somewhere else. Similarly, make sure "Launch:" points to the same EXE. Picture of dxwnd for reference:

Press "Ok".

3. Now start the game through dxwnd or through Steam. The game should now open in a window.

P.S. If the game still starts in fullscreen, the universal fix is to launch dxwnd as admin.