1. Are you solving for the temperature distribution within the solid body in Fluent, correct?
  2. You are mapping this temperature onto the body in ANSYS mechanical, correct? If so, what time step are you using?
  3. Again, the approach I would take is to be mapping the heat transfer coefficient from the steady state model onto the outer surface of a transient thermal mechanical model. Then you can simply run for whatever length of time until the body reaches the temperature you want.