To be a bit more specific, you essentially got one background color (the brightest one) and three darker tones to draw on top. You can't really draw with the background color.
For alpha transparency you can fake alpha by letting your sprites flicker (visible/invisible) between frames, which was rather common back then.
I think one could even add blending between frames to improve the effect, since the original Gameboy had quite some ghosting going on.
So in some way you can have alpha, but you'll have to fake it, if you want to stay authentic.