No. Goalies hardly ever get 1st round picks in trades, it would take way more than a 4th. Caps prospect shelf is already pretty empty and they need to get younger. He also isn't worth exacerbating the salary cap space headaches.
Not saying that trading the 1st is the right thing for the Caps to do, but Frederik Andersen should be amongst their top targets this offseason. They are fine on the cap, and I'm sure the Caps like 3/4 of the teams in the league would love to get a true #1 workhorse goalie for $1m in salary this year.
The reality -- they have absolutely no idea what they're going to get out of Ilya Samsonov this year. He could be their next Holtby, he could struggle out of the gate (especially coming off the injury, in a season likely to be condensed), and have his confidence absolutely destroyed. The Caps need a guy that would be comfortable under a 60-game workload, and they need him to either have a contract under $4-5m, or be on a short term deal, so that they can give Samsonov a raise next year.
That being said, if you figure that Ovechkin is probably looking at a pay decrease next year, they may be able to shop in the market of a guy like Ben Bishop, Darcy Kuemper, or maybe Marc Andre Fleury if Vegas retains.