Take a James Bus Line or National Transportation Services bus from Punta Gorda. There are about 8 departures per day and 1 or 2 are express runs; the rest are "regular" meaning they stop for passengers anywhere along the route.
Try to get an express bus but be aware that express buses don't usually stop between main destinations; check with the driver when you board. If he says that he'll stop at the zoo, the trip should take about 4 to 5 hours and cost about BZ$25. If you get a regular bus from PG, it might take an hour or so longer but it will be a few dollars less.
If the driver of the express bus says he will NOT make a stop at the zoo, then get off at Belmopan and get any regular bus going to Belize City. Ask the driver to stop at the zoo. The zoo is about 20 miles from Belmopan.