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.
Yes, this can be done. I remember answering such a question a couple of years ago. Please search the forum on my name and also there was a response of Jay from ESRI.