You can add second test to final IF.
But what in case when 0 is in first row of given product?

Sub UpdateTotalQuantities2()
    Dim ws As Worksheet
    Dim lastRow As Long
    Dim currentProduct As String
    Dim minQuantity As Double
    Dim currentQuantity As Double
    Dim i As Long

    ' Set the worksheet
    Set ws = ThisWorkbook.Sheets("Sheet1") ' Change "Sheet1" to your actual sheet name

    ' Find the last row in column B
    lastRow = ws.Cells(ws.Rows.Count, "B").End(xlUp).Row

    ' Initialize variables
    currentProduct = ""
    ' Loop through the rows
    For i = 2 To lastRow
        ' Check if the product name is the same as the previous row
        If ws.Cells(i, 1).Value Like "*Total*" Then
            ' Update the Total row with the calculated totalQuantity
            ws.Cells(i, 2).Value = minQuantity
        ElseIf ws.Cells(i, 1).Value = currentProduct Then
            ' Update total quantity with the minimum value
            currentQuantity = ws.Cells(i, 2).Value
            If currentQuantity < minQuantity and currentQuantity<>0 Then
                minQuantity = currentQuantity
            End If
        Else
            ' If a new product is encountered, update the currentProduct and totalQuantity
            currentProduct = ws.Cells(i, 1).Value
            minQuantity = ws.Cells(i, 2).Value
        End If
    Next i
End Sub