I did it for a bike that was too small for me. I don't remember it being any sort of "messy business" - all I did was replace the stem with something more upright (make sure you know the stem diameter), found bars that would fit, found mountain bike shifters with the same gearing and some brake handles and put it all together. Make sure you grease your stem.

Is this a quill or threadless stem? a threadless stem would be much easier assuming you can unbolt it from the front and don't have to strip off the levers and bar tape just to get the bars out. If it is a quill stem, you will have to do that, and look for dirt drop stems. They are tall and witha short reach. Nitto makes one, as I am sure others do as well. You could also replace with the type of stem a lot of comfort bikes use that allow the rider to change the rise.

If a threadless, yes you should be able to find a stem that would meet your needs.