Hmm, disagree with this ruling. You're conflating the first part of Assassinate with the second part.
You get advantage against any creature that hasn't taken a turn. So, in the case you outline, the assassin would get advantage with the attack that triggers combat (assuming that he doesn't do something else before actually firing that would alert enemies and trigger combat, say crashing through a skylight, or dropping down, or whatever). Since the target is surprised, the assassin auto-crits. Initiative is rolled when combat is triggered, possibly by the impact of the assassin's attack. Now, the assassin wins initiative, so again before the target takes a turn. Advantage. However, unless you rule that the target is still surprised (meaning that they have no way to know about their attacker until the attacker takes an action on his or her own turn), the assassin does not auto-crit. They do get sneak attack.
In this situation, I would say that the target is not surprised? Why? Well, first of all, it doesn't make sense from a balance perspective. Honestly, that's my first reason. I say, "really? that seems a bit much." And I review the situation, and I say, "right. The combat started with the arrow hitting the target. So the target knows about his attackers before combat, because he has already been hit by an arrow, an action that happened outside of combat."
Surprise is all about what happens over the course of the 6-second turn. Because, remember, these 6 second turns are all happening simultaneously. So, while one character is charging into battle, another character is drawing a bow, and another character is raising a shield and another character is taking a swipe at a passing foe and another character is quickly casting Shield to block an incoming arrow. All happening at the same time. So, if I'm surprised, I'm not doing much during those 6 seconds, because the actions which surprise me also happened during those 6 seconds.