There is free software called QGIS or GRASS that can be used instead of ArcGIS and has similar features, just not as easy to use.
There is also a website called http://spatialreference.org/ where you can type in the coordinate system the DEM data is in and download a *.prj file (projection file) that HEC-RAS Mapper can read. The USGS National Map website should contain the projection that the DEM is in.
If the DEM does not overlay with the HEC-RAS model, you will need to convert the DEM to the correct coordinate system using software.
If the model only covers the 2D area, the model should be able to create the 2D volume curve data. If you turn off floodplain mapping it may or may not run, I have not tested this yet.
In Mapper you can actually create a surface from either the entire cross-sections or just from the channel (between banks). Once this is created you can than use the cross-section/channel surface and DEM to create one surface in Mapper.