Here is a macro based on your screenshot. If the real layout is different, you'll have to modify the code.
Sub ExtractData()
Dim s As Long
Dim m As Long
Dim t As Long
Dim c As Long
Dim dtmDate As Date
Dim strName As String
Application.ScreenUpdating = False
' Get the date
dtmDate = Cells(4, 8).Value
' Find the corresponding column
For c = 3 To 5
If Cells(3, c).Value = dtmDate Then Exit For
Next c
' Get the name
strName = Cells(5, 8).Value
' Clear the target range
Range("J5:K" & Rows.Count).ClearContents
' Initialize the target row
t = 4
' Get the last source row
m = Cells(Rows.Count, 1).End(xlUp).Row
' Loop through the source rows
For s = 5 To m
' If the name matches ...
If Cells(s, 1).Value = strName Then
' ... go to the next target row
t = t + 1
' and copy the project and hours
Cells(t, 10).Value = Cells(s, 2).Value
Cells(t, 11).Value = Cells(s, c).Value
End If
Next s
Application.ScreenUpdating = True
End Sub