Who cares if he shot first? Han Solo can barely hit a thing! The queen is a better shot, for goodness sake.
Troels Hundtofte analyzed all of the blaster shots in Star Wars to see who had the best aim. Unsurprisingly, Finn came out on top, followed by Padme, and poor, poor Han Solo came in dead last. In Han's defense, he shot a total of 45 times, whereas bullseye Finn only attempted 4 shots. Still, for being an ace smuggler, that's a pretty weak showing.