Lot of people start the TransAm in Yorktown in April. Yes, May is better, but April is workable. It might be pretty cold and wet in the Appalachians, but it won't kill you. It will also likely to be cold in the Rockies, and there's a chance you'll have to wait out a snowstorm to cross Hoosier Pass. And there's a pretty good chance McKenzie Pass in Oregon will be closed, but there's an alternative.
Lots of people fly into Washington DC to do the TransAm. There are a wide variety of options to getting to the start of the TransAm, including the train to somewhere close, one-way car rental, or just riding your bicycle there. If you take Amtrak, just be sure to pay attention to where the baggage stops are, as they won't let you unload your bike at all the stops. The closest airport to Yorktown is Newport News, which is easy riding distance from Yorktown. That's where I flew into, and the price was quite reasonable for me (coming from within the US--coming from overseas likely makes it much more expensive).
As Pat says, if you fly into Norfolk, you won't be able to ride your bike directly to Yorktown (the tunnel doesn't allow bikes), but you can ride to Jamestown and backtrack (or not). There are some friendly folks in Norfolk, however, who frequently give cyclists a ride to Yorktown.