What is the cure time for shotcrete before paint can be sprayed? And what type of paint (latex or oil-based) would work best?

Shotcrete is a placement method for concrete. Thus, cure time is the same as what the coating manufacturer recommends for new concrete surfaces. Because moisture in the concrete surface is part of the concern of bond from manufacturers, a water-based latex may be preferable to oil-based, but you should discuss the appropriate time and materials with your coating supplier.