Hello OP,
I was in Jerusalem just a few weeks ago and felt completely safe.
For the Old City, there's plenty of tour guides available. You can also take a walk along the top of the wall. Perhaps your hotel concierge can arrange a tour. Have plenty of dollars and shekels with you because sometimes the tour guides prefer to deal in dollars. Even the ATMs dispense cash in dollars -- which is great.
The Old City is not for anyone physically disabled because the streets are narrow and narrow.
There's only a few streets where cars can travel.
But for safety and tourism: Jerusalem is terrific.
I hope you have a great trip.
David