Been riding in NYC for 10 years now. Most of those years; I commuted with a roadbike.
2009 Fuji Newest 1.0, 2013 Scott CR1 Pro, 2014 BMC Gran Fondo. 16.5 miles each way.
In the winter, in the rain, in the summer - no problems. For a US$500 roadbike budget;
you'll probably looking at used bikes.