yes, sliding the shaft back and forth (along its axis, the long ways) should be near none. If there is play, the thrust washer needs to be replaced IMMEDIATELY.
All non-ball bearing turbos will have radial play, even brand new. I have even seen turbos that if you push the shaft sideways (light force, from center towards the outside), you can get the wheel to scrape the housing, but they never fail and run for years. Oil pressure keeps the bearings centered when idling, and aerodynamic pressures keep the wheels centered (more or less) when under boost. The bore the bearings ride in is actually egg-shaped just slighty, so at it rotates it "wedges" oil in between.