1) If you only have 256Mb of RAM then Ubuntu will run, albeit more slowly and you will not be able to use fancy desktop effects. According to Wikipedia Ubuntu needs 256 Mb of RAM - but 384Mb is recommended. If you can, increase the amount of RAM in the computer.
2) As saikee says Linux has generic drivers. If your graphics driver is Nvidia or ATI, or some others, you will be able to install propiatary linux drivers.
3) For the wireless driver have a look at e.g. "Belkin Wireless G router" at
http://ubuntuforums.org/showthread.php?t=680550
4) If you can't increase the RAM then there are other derivatives of Ubuntu you can use e.g. Xubuntu (Xfce) or Crunchbang (OpenBox) which need less resources (about 192 Mb of RAM). They will certainly run quicker on the amount you have...
5) Another thing, if you want one of the *buntus, download the Alternate Disk (straight install) rather than the Desktop Disk (LiveCD) because the first only needs 64Mb to install but the second uses LOTS more.