It depends on what you mean by "nearest arterial road". If you mean "nearest" in terms of straight-line distance, then you do not need Network Analyst, and you can likely do this using a host of GP tools. If you mean "nearest" in terms of driving distance (or time) along the road network, then you would need Network Analyst.
Using Network Analyst, you can create a new Closest Facility analysis layer and load your "points" (e.g., Point A, in your original post) as Incidents. Then load the vertices of your arterial road features as Facilities. Solve it to find the closest single Facility (i.e., the closest point on your arterial roads). Next, to get your path from a point to an urban center, create a Route analysis layer, and load your point as the first stop and your urban center as your second stop. Then solve to get the path. Finally, buffer this path as you wish.
If this is all new to you, I would suggest you work through the Network Analyst tutorial, which should give you the information you need to get started. Let me know if you have any further questions or problems with the workflow suggested above (or if I somehow misunderstood you).
