Get yourself a sturdy umbrella. Wear whatever hat you want. In colder wet situations I recommend a fleece or wool beanie of some sort. But as you likely know there are wind tunnels between the skyscrapers which will make the rain and snow go sideways. You can use the umbrella as a bit of a shield so it doesn’t hit your face from that sideways action. I used to commute daily to Manhattan and this was my system. Always have an umbrella with you in the city. Pair it with a long coat, at least 3/4 length for more body coverage.