Forget the fact that Ullmark is a 22-year old rookie. There's almost nothing he could have done differently on the second goal. His "flopping" issues have been vastly overrated and he's getting better at that issue by the game.
But do you want to know why he looks like he's "flopping"? Because he plays at the top of his crease and doesn't give up shots from distance (cf Chad Johnson). But, more importantly, he also moves very well on his feet. That means he's set for shots that other goalies are sliding into, which means he's less likely to give up a rebound in the first place. Every shot that hit him tonight got swallowed up.
That's my favorite part of his game. He's ready for every shot - well, except the ones that bounce off his defenseman right to the stick of an opposing forward on the other side of the net and then get knocked back into said forward, bouncing back across the net...again...and rolling off the defenseman into the opposite side of the net. He's not set for those. He flopped in an attempt to make that save. I'll cede that point.