I found a history book (A History of the Christian Church, Revised Edition by Williston Walker) at my local used book store. It was copyrighted in 1959. Have there been any developments in research since then (new archaeological discoveries, etc) that would be significant enough to motivate me to find a more recent book? How concerned should I be about error or incompleteness in an older history book?
If you are serious about learning the history of Christianity, you should be motivated to find more books period. A single book, especially one attempting to cover a massive subject like Christianity, cannot possibly suffice for anything beyond a cursory read. It will be "incomplete" regardless of how old or new it is, if only because you're fitting thousands of years of history into a single volume.
That said, A History of the Christian Church is a classic that would be 100 years old in four years, and copies are still sold. The author, Williston Walker, was a distinguished theologian at Yale. I would argue there's nothing wrong with reading the book, considering you already have it anyway. You can compare it to later works and see how the perspectives have changed.
As for trusting it and concern about errors: just be aware that not all of this 1959 edition's information will be the most up to date. So if you find conflicting information, you can assess the book's correctness for yourself.
Books on the history of Christianity are inevitably influenced by the author’s own religious prejudices. In my opinion the most objective books are those from the end of the 19th century by writers like Wellhausen (Old Testament) and Harnack (early Christianity). At least they treated the subject as a serious historical discipline.