get index of DataTable column with name

You can use DataColumn.Ordinal to get the index of the column in the DataTable. So if you need the next column as mentioned use Column.Ordinal + 1:

row[row.Table.Columns["ColumnName"].Ordinal + 1] = someOtherValue;

Warning:

This code returns the next column, so the one after ColumnName, as requested in the question.