I believe a common approach is to use a linear activation function for regression problems unless target variable has certain semantics that suggest the use of other non-linearities (sigmoid, tanh etc.). Also consider rescaling your targets instead of trying to match the desired output with activation functions.

From you description (I might be wrong though), it seems like the 0 output is a special case. In this case you might want to use a binary classifier to classify input samples into two classes first. For class 0 the output is 0. For class 1 you use another model (regressor) that outputs a prediction.