Here are a few bike rental shops in the Jerusalem area.
My perosonal recommendation is the Bilu Bike Shop in Katamon. Aviram (owner) and all the team are real pros and great to work with. See Bilu Bike Website
We have no personal experience of the following suppliers – please give us feedback.
Both providers encourage you to phone and reserve in advance.
In Moshav Aminadav & sometimes at Sataf
Bike Shop in the Katamonim area – Moti Bikes