Yea...having an additional step to update a table in which you can set the value in the inital insert is probably an extra, avoidable process. Do you have access to the original insert statement where you can actually just insert the part_description into the part_description_upper column using UPPER(part_description) value?
After thinking, you probably don't have access as you would have probably done that so should also give some options as well...
1) Depends on the need for this part_description_upper column, if just for "viewing" then can just use the returned part_description value and "ToUpper()" it (depending on programming language).
2) If want to avoid "realtime" processing, can just create a sql job to go through your values once a day during low traffic periods and update that column to the UPPER part_description value for any that are currently not set.
3) go with your trigger (and watch for recursion as others have mentioned)...
HTH
Dave