I just got an old mare that is sway back.
I was wondering is there's anything that can do to improve it.
And if light occasional riding would hurt her.
Also what could I do to keep the saddle from bridging?
Doing belly lifts with her (running your thumb firmly down her mid-line under her belly until she flexes her stomach and lifts her back) can help to strengthen her stomach muscles which can help to prevent further swaying (and maybe even correct it to some degree, depending on why it's swayed). A swayback pad that is properly shimmed can stop the saddle from bridging, but like others said, the severity of it will dictate whether riding is wise or not.