Failure ID on hard drive 1 (usually) indicates a SMART drive failure warning - so the drive likely requires replacement as you suggest. Before you do that though, it's worth checking that the connector which connects the drive to the board is properly in place and connected as if it isn't it could be causing a failure.
To answer your direct question, it is possible to copy the HP factory recovery partition over to another drive, but you'd be best advised to use larger than a 160GB as you'd not be left with much after you'd done it. You can get a 1GB spindle drive for very little and this would be a better way to go. Do you have access to another computer? If so, you could check to see if you can actually read that partitions on the drive. If you can, and you can afford a replacement HDD then you might want to look at getting a USB drive duplicator (check Amazon). The reason I suggest this is that if the drive just has a SMART failure but is otherwise (currently) readable then you might get away with a straight clone (onto a drive with the same capacity) and get up and running that way.