You are a short distance from public transportation. Go down the hill on Impasse du Mont Choisy, make a left and go to Rue de Happy Bay, turn right and walk a very short distance to Rout de la Savane, which is the main drag between Marigot and Grand Case. (This is how your driver from the airport will get you to your villa. Be careful! If you want to walk to Grand Case turn left. Walk to where the road makes a "Y". To the left is an international "Do Not Enter" sign. Turn left and you are in beautiful downtown Grand Case on Boulevard Leonel Bertin Maurice. On your way back you can walk westward on the beach past where you turned off, and stay on the beach until you pass the cemetery (cimetiere in French). Take a left and walk across the field a short distance and you are back on Rue de Happy Bay. I would do this only during the day. It would be very dark at night and a very good flashlight would be a must. You could also catch a bus from the main drag to downtown. All buses are privately owned vehicles, most of them look like VW buses. If it is a Dutch registered one the number plate will start with "BUS". On the inside of the windshield will be a sign saying the furthest distance it goes. Wave it down. There are bus stops but they will stop anywhere. When you get to your destination say "Please stop the bus." I is always good manners on the Island to say "good day" upon meeting someone, here the driver and fellow riders and " thank you, goodbye." The fare is $1.75 for the entire rout, over 65 free. When in doubt tell the driver where you want to go and he/she will tell you when you are there. Again politeness is key. Smiles beget smiles. Blank looks beget frowns. It's just the way things are. Remember this and your island experience will increase many fold.
To get to Marigot go right on the main road. Do not attempt it on foot unless your are an Olympic class mountain goat! Take a bus using the above instructions. Again day time only. I'M not sure about getting to Philipsburg, but it would be to the left on the main drag. If you see a windshield sign saying "Phillipsburg" you are in luck. If not flag it down, and after saying the magic words, "good day, how are you" ask the driver about bus service to Phillipsburg.
My final bit of advice is that when you leave the baggage area and past the glass doors is look across the hall and to your left across from the food court, and you will see "Dutch Delight", a small convenience store, go in, get a cold drink, a "Daily Herald" (A remarkably good locally published, English Language newspaper), and map of Saint-Maarten, Sint Maarten by Michal Kasprowski. The drink will cost $2-3 the paper $.75 and the map about $15. The map price might seem a bit high, but let me tell you it is worth every penny. You will be able to easily be able to follow my instructions on it and use to plan your outings. Remember to use the "magic words" at the cashier and you will be off to a great start on your island adventure. Have a great time and don't forget pictures and a trip report.