I run them all (FO1, 2 and tactics) on windows 7 Home premium 32bit with no problems at all!
What I did was a Manual Installation of Fallout 1/2: http://www.nma-fallout.com/article.php?id=1593
It's basicly copying and pasting of the game data from the CD to your HD. Then install it as a Humongous Installation (see link). So you don't need the CD anymore.
Patch it only with the official patches and the High resolution patch (Turn resolution to 16bit in game menu and the screen to 800X600)
Normally it should be working properly.
Troubleshooting FO1&2 installation: http://www.nma-fallout.com/forum/viewtopic.php?t=5061
Patches: http://www.nma-fallout.com/forum/dload.php
Oh and maybe the screen refresh patch will also be helpfull.
Edit: Sorry I can't find the source for the refresh patch anymore. It's a French fan made patch, figures!
So, I uploaded it for you: http://www.megaupload.com/?d=0BW7LTRU