I follow this link (Modifying field properties—ArcGIS Help | ArcGIS Desktop )
I do not find "the corresponding Data Type drop-down list" in my feature class' field property
What type is your data (Shapefile/FGDB Feature Class/Enterprise GDB Feature Class)?
Ensure you are performing the activity in ArcCatalog Window/Dialog.
Is the Field Empty (No Values)?
What is the current datatype of the field, and what is your desired datatype for the field?
If the field contains data, you cannot change data types that would lead to loss of data. For example, you cannot change a long integer or float to a short integer, or change a blob to a text.