I'm sure some software will need to be re-written between now and 2038, but I don't think it will be quite as bad as Y2K just because that was only a 15 year gap (Sometimes less), whereas this is over 24 years.
I just think a lot of software will be naturally replaced between now and then. And while there will be a slight mad scramble to fix stuff at the last minute, I don't think it is Y2K-2.