I wonder if it is possible and how much it would cost to travel from Washington DC to Newport News (or somewhere near) by boat along the Potomac River?
There is no long-distance water transportation to or from Washington, D.C. The nearest cruise terminal is Cruise Baltimore. Similarly, there is no long-distance commercial passenger water transportation from Newport News; the nearest terminal is Cruise Virginia, and the last major cruise line, Carnival, left in 2013. As far as I can tell, there are no services between Baltimore and Norfolk, either; the vast majority of passenger boat traffic is headed for Florida or the Bahamas.
Considering how ridiculous I-95 traffic is, especially now with the massive 95 Express Lanes construction project underway, you'd think a boat service would be commercially feasible, but as of now, your best road alternative is probably Amtrak. There is Northeast Corridor service connecting Washington to Newport News two or three times a day, and daily service from Washington to Norfolk.
Incidentally, Washington has no commuter ferries, either, though such services have been proposed and studied for decades. The nearest ferry crossing is White's Ferry, about an hour northwest of downtown in light traffic.
Locally, there are a handful of water taxi services and dinner cruise companies connecting certain points in and close to the city during tourist season. Not all companies serve all locations, and the hours of each operation are also highly variable. For example, water taxis to Nationals Park operate only when there are baseball games or major concerts. The stops include
- Alexandria, Virginia waterfront (aka Old Town)
- Georgetown waterfront (31st & K St NW, D.C., at the Washington Harbour complex)
- National Harbor (Prince Georges County, Maryland)
- National Mall (West Basin and Ohio Drives NW, D.C.).
- Nationals Park (aka Diamond Teague Park, Capitol Waterfront, South East Waterfront)
- Southwest Waterfront (D.C., inc. The Wharf)