Sorry, I don't have a good answer to your question.
However, I do not think it is a network analyst issue. Or, at least, there is not a Network Analyst way of getting the answer you want.
You might have some luck trying to find out how to get the azimuth of two points along a line. You do one point as the location of your vehicle, then use the curb approach to figure out which direction along the line to go to put the second point.
