What is the manga where to escape hell the person has to do a bunch of good deeds?
It's Togari. It's about a boy named Tobei who has been dead and living in hell for over 300 years.



Summary:

He tried to escape hell multiple times but got caught repeatedly. One day he was made an offer that if he could kill 108 demons in 108 days, then he would be free to return to Earth.

He agrees on doing this and then gets sent into modern-day Japan.

Things that definitely match your description:


On the cover you can see his magical sword that can easily be mistaken with a wooden sword or stick.
He lives together with a girl name Itsuki Asagi. He saved her from getting raped and murdered by a thug. After this, they live together with her grandfather.
In volume 3 they are caught in the middle of a bank robbery with one
of the bandits holding Itsuki hostage.