The years immediately prior to WW2 witnessed a major Palestinian revolt against the British occupation and the increased Jewish immigration: http://en.wikipedia.org/wiki/1936%E2%80%9339_Arab_revolt_in_Palestine

The British army used brutal collective punishment against rebelling villages, and armed Jewish militias to counter the Palestinian ones. The Irgun was also arming itself towards the end and fighting virtually everyone.

Jerusalem itself was a scene of major fighting.

There had also been earlier incidents in the decade prior: http://en.wikipedia.org/wiki/1929_Palestine_riots

So the basic answer is no. Not a very nice place to live. The Jewish population was significant, but the old city was (and still is if you visit today) very much Arab in character.

TL;DR it was a war zone.