This game runs fine off of 512. Ive got it runnin on two different machines, both have 512m. One is running a radeon 7500 at 1024x768 the other a radeon 8500 at 1280x1024. Both run the game perfectly. If your playin a single player game at full res with 64 bots going, then yes the more memory the better. But it will still work. Lots of people have been having problems with this game though, and its usually sound related.
Check your sound settings, if hardware acceleration is on, turn it off. If its off turn it on. Lower the sound settings. Check the bf1942 forums, this problem has been answered many times on there.
Alot of people running fast machines have been crashing to the desktop. The only time it happened to me was when I used a no cd crack. If your using one get rid of it, they just cause problems with this game.
Try reinstalling the game. Just be sure to delete the remaining BF1942 file from your program files and clean your registry before you reinstall. Reinstalling has also had its issues with this game.
And last but not least, install in this order: bf1942- then RTR (if you have it)- then the 1.31 patch. Try it to make sure it works and then add any mods like Desert Combat last.