When I was there for work a few times several years ago I basically used my old screw mount S-M-C 28mm f/3.5, S-M-C 55mm f/1.8, and vivitar 200mm f/3.5 lenses almost exclusively. I did get a shot of Dome of the Rock from on top of Mount of Olives and used my shitty 2x telephoto converter with the 200mm lens which produced reasonable results but other than that one time I didn't use anything larger than 200mm. This was with my Spotmatic F so full frame. There were only a handful of shots where I really wanted to go wider than 28mm mostly when I went to Caesarea and looking back wish I had my 17mm fisheye then. I was in Haifa for work so know that city better than others and tried to make as much use of my free time as I could.
Outside it is bright, like really bright if you are not use to it there so a slower lens works great but inside it is always dark as they have tiny windows there. Stopping a lens down to get better IQ isn't a concern when outside.
Some places worth going to that aren't on your list, it is a small country so it doesn't take long to get places, are the Baha'i Gardens in Acco and Haifa, I would also suggest doing the Stella Maris Monastery (Cave of Elijah) in Haifa. Since you are doing the old city there is a lot to see and do there and expect to go through metal detectors from time to time. The church of the Holy Sepulchre will be packed all the time and try to steer clear of the the people trying to sell you an olive branch near the Garden of Gethsemane.
Of note if you are staying at one of the Dan hotels they often have free guided tours that you can sign up for at the front desk. The tours are on Friday or Saturdays (I forget which) and they spend half to 3/4 of a day on them and they have a set of various ones they rotate through. As an added bonus they are usually pretty small groups and not rushed.
When I was there a few of us paid for a private tour of the old city. It worked out to $140/person but we got picked up at our hotel at 8 AM and didn't get back until 9 PM and got to go at our own pace and see what we wanted to. One guy was a very devout Catholic so we did the stations of the cross, all of them, even the non catholic ones. We saw most of the major sites and got a lot out of it. The next day is when the very devout Catholic and I went down to the Stella Maris as he wanted to get the olive wood cross he bought and touched to all the holy sites and stations of the cross blessed by the head brother there we ended up getting a luck break private tour by him as it was a slow quiet day and we happened to be there at just the right time. Haifa, Acco, and Caesarea are all fairly close to Tel-Aviv so doing one of those one of the afternoons would be doable.
Other than that, my only advise is to drink from the firehose of life while there. Eat falafel, shawarma, tabouli, and kebabs (different from shish-kebabs), and if you want a good burger stop in at a Moses (
their logo looks like this).