The book is pretty old, yes, and many of the topics are not uptodate any more (I just cannot believe how fast genetic research has developed in the last 20 years and the Internet was not even there when the book was written). But then - GEB never was a book for the classroom to teach computer science. The original intention, as confirmed by the author in the preface of a german anniversary edition of GEB in 2015, was to provide a graspable access to Gödels proof for a general audience. The project then escalated from an intended essay to the epic work it is.
I read it at the age of 19, multiple times, and for me it was the ultimate primer for everything: whatever you deal with, take it apart, change the context in which you are looking at it, look from the distance, look close and from all sides, extract patterns and apply them somewhere else, combine, prescind, generalize, play, be curious about each and everything and then - while reading the book, and on other occasions in life - enjoy brief moments of epiphany. And for this purpose - to open the mind for another perception of everything - GEB is timeless.
I gave it as a gift to a colleague at work, with whom I regularly ended up in funny, crazy scientific discourses. And I keep two shrink-wrapped copies from 1989 for each of my two kids - if I am gone and the kids ever want to know how I perceived the world, they just have to break the seal and read.