With only 2 days to explore Fredericton and Saint John, I fear you will spreading yourself too thin having to stick to a strict timeline of getting to St. Andrews.
Being a New Brunswicker and travelled all over the province, I would recommend getting off the plane and heading straight to Grand Manan Island for a quiet getaway. It happens to be very convenient for your time in St. Andrews afterwards.
If you do insist on touring Fredericton & Saint John, hit up both the Farmer Markets. The Boyce Farmers Market (Saturday only) in Freddy and the City Market (All but Sunday) in Saint John.
Around Fredericton; Boyce Market,
Kings Landing, Picaroons Brewery, New Brunswick Sports Hall of Fame.
Around Saint John; City Market, Reversing Falls, Martello Tower, St. Martins.
You can't go wrong in NB, in my opinion.