How can I add the river run time as a field in my target table?

First, in the Column Mapping step of your river, add a new field by clicking ‘Add Field’ at the bottom of the page.

Then, in the ‘expression’ column of your new field, enter the expression for getting the current date/time in the native SQL dialect of your target database. The below example is for a river with Snowflake as the target:

The expression gets cut off in the above picture, but it is CURRENT_TIMESTAMP().