The best local park I know near the airport is Pasir Ris Park, where you can find a good number of local "garden" (sunbirds, woodpeckers, triller, mynas, barbets) and mangrove species (fantail, hawk, kingfishers, herons, egrets, re-introduced red junglefowl), as well as the odd rarity (that season is good for migrants like cuckoos and some shorebirds can sometimes be found near the seafront, as well as hornbills). You need a cab to go there and back, considering the amount of time you have.
Near Changi itself, you may have other good birding opportunities (I heard Malaysian Plover and Nightjars may be found near the airport), but I don't know where exactly, and it will be for just a few species.