API calls, when being ported to VB.NET, often have to be changed. Wherever you see anything declared as a Long, you should change it to Integer. Often, when these Integers are handles to things, they should be IntPtr instead.
As a rough guess, I have changed your API declarations and code to be as follows:
Public Declare Function FindWindow Lib "user32" Alias "FindWindowA" (ByVal lpClassName As String, ByVal lpWindowName As String) As IntPtr
Private Declare Function SendMessage Lib "user32" Alias "SendMessageA" (ByVal hwnd As IntPtr, ByVal wMsg As Integer, ByVal wParam As Integer, ByVal lParam As Integer) As Integer
Dim hwndWinamp As IntPtr
Dim asdf As Integer
Private Sub Timer1_Timer()
hwndWinamp = FindWindow("Winamp v1.x", vbNullString)
asdf = SendMessage(hwndWinamp, WM_USER, 0, 0)
TextBox1.Text = asdf.ToString()
End Sub
I've also applied what Merrion said. I assume you have WM_USER declared somewhere with the correct value. If it still doesn't work, make sure you have Option Explicit AND Option Strict turned on, and tell us what error the compiler gives.