According to documentation, the CS_DROPDOWN style is available for top-level windows only, not for controls.
You can simulate simple shadow using an additional empty panel. For example, if your panel is called panel1, you can insert another one programmatically:
auto shadow = gcnew Panel;
shadow->Size = panel1->Size;
shadow->Left = panel1->Left + 4;
shadow->Top = panel1->Top + 4;
shadow->BackColor = Color::FromArgb( 0xAAD3D3D3 );
shadow->BorderStyle = BorderStyle::None;
panel1->Parent->Controls->Add( shadow );
shadow->SendToBack( );