If when authoring a map document in ArcMap relative paths were used then you could just replicate the location of the data on the Linux machine. This is a common practice for distributing .pmf files.
So, if you want to distribute data together with the published document, you need to use relative paths when you create the map (use the File > Map Document Properties menu). Then, thanks to using relative paths, the published map file will be able to access the data as long as it stays in the same location relative to the GIS datasets.
However, since you hadn't used relative paths, the only way to manage the data paths with respect to Linux paths, is to re-publish the map document in ArcMap with Publisher extension enabled and set relative paths for the map document beforehand. Alternatively, you could also use absolute paths, but make sure you use the same path you expect to have on the Linux machine.
Esri help page: About managing data sources for ArcGIS Publisher