Set Snapping to: Snap Base: Active / Snap Target: Face / Align Rotation to Target, and enable Affect > Move. Select your object, hold CTRL near the target surface to snap it by its origin. For accurate base snapping, move the origin to the base (Options > Affect Only > Origins) and invert either the object or its origin on the local Z-axis (S > Z > Z > -1). Don't forget Applying Scale if you encounter any bugs.