If you intend to do routing on a Network Dataset in ArcGIS for Desktop using OSM data, you should really have a look at ESRI's Open Source and free "ArcGIS Editor for OpenStreetMap". It contains specially designed tools to create a routable Network Dataset from OpenStreetMap data, that deals with the specifics of OSM data in this area.
ArcGIS Editor for OpenStreetMap:
To be honest, I have never used the routing related stuff, so can't say how well it works, but I have used other parts of the same toolbox (unrelated to routing), like the *.osm file import.
Please note (I am not an ESRI employee but have been in contact with some of the ESRI developers via the Github site), that ESRI is working on a new release of the toolset, that contains some major fixes for OSM multipolygon relations and the processing of these in the toolset, fixing some issues I and other people reported to ESRI. This will have a major effect on the quality of polygon output by the tool. I can't say when it will be released though.
The Github repository for the source code of the tool is here: