Immediately I am not able to think of a formula solution. However you can use an event code
A1 down and B1 down has data. C1 has validation of list of column A data.
after copying the event code below at the appropriate place when you choose a value ofr C1 FROM VALIDATION then automatically corresponding B value will come up in D1.
right click the sheet tab and click view code. in that window that comes up copy paste this event code
Private Sub Worksheet_Change(ByVal Target As Range)
Dim cfind As Range, x As Variant
If Target.Address <> "$C$1" Then Exit Sub
x = Target.Value
Set cfind = Columns("A:A").Cells.Find(what:=x, lookat:=xlWhole, LookIn:=xlValues)
Target.Offset(0, 1) = cfind.Offset(0, 1)