a few ideas, not in any particular order:

just take a drive around the island...but fyi, if you're one of those that thinks driving on SXM is stressful, you're in for a surprise on SBH...some very narrow cork screwing roads that wind up/down some pretty steep hills...rent a SMALL car, and beware speed bumps

check out the overlook in Colombier...excellent high up panoramic ocean views, even has a map showing you what you're looking at...down below you can see the cove of Colombier, one of our favorite beaches, but it's an offroad hike so save the beach for another visit

go to Nikki Beach, famous bar/restaurant in St. Jean (east side of Eden Rock)...you might even see some celebs and billionaires partying and dancing on the tables, especially if you go on a Sunday afternoon

go up to the lighthouse above Gustavia...awesome views of the town and marina...especially if the harbor is full of mega yachts

in Gustavia, have a burger and a beer and/or Ti' Punch at Le Select...although according to JB himself, not his actual inspiration for "Cheeseburger in Paradise", but he did spend a lot of time there and was very good friends with the original owner, Marius, who passed away in 2020 at 97 y/o..."dive bar" with a lot of character and cool pics on the walls, (although dive bar in SBH is an oxymoron)

right in Gustavia is Shell Beach...might be my favorite beach...beautiful small beach in a very picturesque cove...great for people watching and you can have a drink at the bar/restaurant Shellona right there

from the roundabout above the airport, watch planes pass just over your head as they land