If it's a real old monitor, it may have trouble making a resolution change from the "Windows Loading" screen to the desktop. Once you get your blank screen, turn off the monitor and turn it back on. If you still don't get a picture...
Go into Safe Mode. Set the resolution to 640x480 at something like 256 colors. Then, in the display properties, choose Advanced, choose the "Adapter" tab, the click the botton "List All Modes." Choose one that says 640x480, 256 colors, and 60 hertz (or whatever the lowest number displayed). Then shut down the computer, shut off the monitor, and then turn everything back on. If you still get the blank screen after the "Windows Loading" screen, turn off the monitor and turn it back on. If there's still no go, it could be a video card incompatibility problem with XP.
A good experiment if you have the capability woud be to attach the monitor on the "junky" computer to a newer one and seeing if the monitor behaves ok. If it works ok, you can then start seeing what display modes the monitor will run. If the monitor still doesn't work, it could be a bad monitor. If you know the monitor works on a newer machine at a high resolution (like, 1024x768) at high refresh rates (like 80 or 85 hertz) then it's likely the hardware on the "junky" computer, whether it be a windows XP incompatibility problem or something else.
If the computer worked fine running Win98 at different resolutions, it almost definately is an XP compatibility problem with your video card (probably not the monitor). You could always try to get a new video card, if you think it's work spending money on the computer.