So what does Solomon have to say about authority?
Obey it (v2)... Solomon is clear about this. God expects us to follow and obey the people who have authority over us (unless they are expecting us to sin against him). Sometimes it's not that simple because we might be asked to do something that seems wrong when there's a possibility it could be ok under circumstances we're not aware of. Sometimes we need to know that the authority might know better than us at the time and the best thing to do is trust them but even if we can't, we needn't worry because God is still the ultimate authority in charge.
Work with it (v3)... Don't fear those in authority over you. They're just people like you and me. That's why Solomon doesn't want people to rush away from him. Why should they? He's only trying to do the right thing for people before God and to be the wisest king he can be. And even if he wasn't, they needn't worry because God is still the ultimate authority in charge.
Respect it (v4)... It's usually possible to question authority to try and understand something better or to resolve something we disagree about but even when it isn't, we are expected to respect authority anyway. And if we can ask questions, let's do so respectfully and with appreciation of the fact that we don't have the same perspective as those in authority over us. We don't need to worry because God is still the ultimate authority in charge.
I think the main message Solomon wants us to realise is that God really is in charge over any of the authorities we experience even when they get it wrong, or have wrong motives, or if they don't give us opportunity to ask questions. God still orders events and makes decisions about what to allow. He is the one who has ultimate power and controls the things we can't (like the wind and the future).
It's not always easy to deal with authority but it is always possible to be right with God and have peace because he is the ultimate parent/boss/teacher/rule/law/country/government. Ask God to help you to respond well to authority and also to help you if you have authority over others.