Where will my bus be at the Port Authority Bus Terminal in New York?

Your Greyhound will certainly leave from Port Authority in New York, some tickets don't have the door or gate number printed on them, but everything is available at the station. I've done this myself on a few occasions, it's a big station and they've organized it a little like an airport in the sense where they have some panels up with the next departures and their corresponding gates.

Worst case there is always an agent there, either at the ticket counter or by any of the gates, you can ask them they know their routes very well. Just make sure to get there well ahead of time, it's a huge and very confusing place

Enter at 42nd and 8th. It's the big building with the taxis and sketchy characters standing outside (or if arriving by subway, enter through the direct subway entrance). Go downstairs to gates ~60-90ish. There are signs labeling it, but I don't recall the exact range listed. After descending the stairs to the correct level, the bus to DC will be somewhere ahead and/or towards the right. If I recall correctly, it should be Gate #86 or so, but there will be a sign somewhere straight ahead in the vicinity of the stairs with the full listing of Greyhound destinations and gates.

You can find your gate by asking someone from the Greyhound ticket booth or you can use the new touch screen that tells you your gate by searching for your travel id located somewhere on your ticket.