Wondering what the best options are here, get to the bus station and bus it from there, or get private transport right from the airport? Or something I haven't thought of?
You can get into town (perhaps taxi is the only option, though that seems unlikely) and take a public bus. Check here for schedules, but my philosophy about Balkan buses is that the only schedule that counts is the one posted in the bus station.
There are two transfer companies. I cannot vouch for either one of them.
I have a strong suspicion that you could simply walk to the taxi stand at the airport and find a driver willing to take you to Kotor, but I have never done that anywhere (I'm too cheap). The distance seems to be about 55 miles, but I think you will encounter some curvy roads, which will make the trip longer than one might expect and likely costlier. Montenegro is beautiful, and that nearly always means curves.
Great thoughts, I’ll keep all of that in mind! Any idea what a taxi right to Kotor would cost?
I have no idea. Perhaps someone else does.
Taxi from Podgorica airport to Kotor costs from 55 to 75 euro, depending on the season.
We took a cab. It wasn't hugely expensive.
That trip alone is worth doing it right, with a guide/car or a private car. I wouldnt rush it.