What do I need to do to finish a roof with exposed rafters on a shed
The diagram you showed is fine. That is really all you need to do. I might add a small strip of 1X2 along the wall as a nailer for the soffit.  The only alternative would be substituting a vented soffit panel for the solid soffit. Using a vented soffit would help a little in exiting hot air and give a bit of ventilation. Best used with a ridge vent for really good ventilation, but better than nothing if used alone.