How can I reduce back pain on a shared bike with poor shock absorbers
I seriously doubt that it's due to the lack of shocks, unless you have some really bad roads there -- many people ride bikes great distances with no shocks, skinny, stiff tires, and no real springs in the seat.  

More likely your problem is with your posture -- seat too low, handlebar too low, handlebar too close or too far away, and/or pedals at the wrong angle to the seat.  

Presumably you can adjust the seat height, so the first thing to try is to move it a bit higher.  When the seat is at the proper height your knees should be nearly straight at the bottom of your stroke.

Next, see if you can adjust the seat forward/backwards.  Being too far either direction can cause back problems, so try several positions.

You probably can't adjust handlebar height, but if you can, try raising it.