How do I command one column to update the date based off the update to another column?

The function NOW() will be recalculated whenever Excel calculates anything, so it won't reflect the date/time that column B changed. You'll need VBA code, so it will work only in the desktop version of Excel for Windows and Mac, and users will have to allow macros.

Right-click the sheet tab.

Select View Code from the context menu.

Copy the code listed below into the worksheet module.

Switch back to Excel.

Save the workbook as a macro-enabled workbook (*.xlsm).

Private Sub Worksheet_Change(ByVal Target As Range)
    Dim rng As Range
    Dim cel As Range
    Set rng = Intersect(Range("B2:B" & Rows.Count), Target)
    If Not rng Is Nothing Then
        Application.ScreenUpdating = False
        Application.EnableEvents = False
        For Each cel In rng
            Select Case cel.Value
                Case "Inprogress", "Sent For QC", "Revision", "Complete"
                    cel.Offset(0, 1).Value = Now
            End Select
        Next cel
        Application.EnableEvents = True
        Application.ScreenUpdating = True
    End If
End Sub