G'morning Cheski!
You ask...
"What would a Christian Missionary be doing in Old Jerusalem?"
Giving Bible studies to Moslems mostly - usually hiding under staircases, or at night in the Jerusalem Forest beside the Yad Vashem. (You must be familiar with the punishment for Moslems caught studying with a Christian and the fate of the Christians who minister to these people?) They were the ones hungry for the Word of God. They sought us out, as did some Bedouins, and so we taught them.
The Jews we met, the practicing ones (most Israeli's are secular), wouldn't lower themselves to speak to us - not that we attempted to interact with them outside of the boundaries of common courtesy, which we didn't find so common amongst that particular group. They couldn't have known that we were Christian, it's not as if we wore tee shirts! And we didn't and don't belong to any group, Cheski, I thought I'd made that abundantly clear. We were in Israel because God sent us there - period.
I have several Jewish friends, some secular, some not, but I take St. Francis of Assisi's advice when people clearly aren't interested in Christ. He said,
'Preach the Gospel daily - if necessary use words.'
Shabbat Shalom - Jesse.