The airport bus is easy, clean, and cheap and brings you right to the center. It should still be operating when you arrive. Alternatively, you can use Uber - it's cheaper than a regular taxi, but don't be surprised if a "normal" taxi comes to pick you up, it tends to be the same guys. You might be able to negotiate a cash price for him to take you back at the end of your trip as well.
The good news is that it's a Mediterranean country, so restaurants will be serving dinner until late, I.e. you don't need to rush too much from the airport!
I'm not sure about the excursions, but at this time of year I think you should be fine to book on the spot. It doesn't hurt to ask if there is a discount for paying in cash, a lot of places still do this although not to the same extent as 10 years ago.
Dubrovnik is definitely the kind of place that is best enjoyed strolling around and discovering things on your own - leave at least a half day if not full day of unstructured time to walk around the old town. Evenings are especially beautiful and you might be lucky to see some traditional live music. A short tour might help you appreciate and understand the history a bit more but I know that's not everyone's thing.
In addition to the great suggestions from pp, there are also day excursions to Montenegro - might be cool to do two countries in one trip! While it's not what I'd usually recommend for a first trip to DBV (especially such a short one), if you want a sense of adventure and don't mind spending more time on a boat getting there and back, it's an option. Honestly, I'd recommend focusing on DBV and maybe one of the islands this time, but the reason I'm putting it out there is because I know I would've found that exciting as a teenager!