Go with a group, that is the easier way. I was there in Jerusalem with my daughter last August just a week after the guy was caught with a holdall of pipe bombs about to get on the JLR. We were there, that time, just as two individuals, and we simply kept away from the big market and some of the other very crowded places in Jerusalem, including the Muslim sector. We were there in Jerusalem for 6 days, and enjoyed it, even with those restrictions.
We had both been previously, as a part of a group covering various parts of Israel, but we wanted to explore Jerusalem on its own instead of doing it as spare time on a country wide tour.