Time Machine question

I have an Imac 2009. The external drive where I would store my backups, stopped working and thus, when my Imac began having trouble and needed to be re-imaged, I could not resort to this external drive. I was using Acronis true Image to perform backups with this external drive.

I also cannot get my Imac to reinstall the O/S from its internal sources, for some reason.

I have a friend who has the same Imac as me and he has an external drive in which he does Time Machine backups.

If I plug his externnal drive in my Imac and load his time machine backup, will it result in my Imac being restored with his O/S and files? If so, it would solve my problem.