Moving from one self-hosted WP install to a different host

I may have gotten myself into a pickle trying to help out a friend. He had a WP blog that was being hosted by a local “web design” firm that was giving him a lot of hassle and unwarrented charges for maintenance, etc. I convinced him to end this arrangement and told him I could help him put the files up on a new WP install on a host of his choosing.

The back up he received doesn’t look like a normal extract from WP (file structure doesn’t look like what I would expect), so I’m not sure I can do an import. Will probably need to ftp files manually to the correct location.

Can I safely save files of the same name from the backup to the fresh WP install on the new host? If not, any suggestions as to how I might go about piecing this back together?