Thanks for writing. If you have pretty decent soil or amend your soil with compost from time to time, you shouldn't need to fertilize your star jasmine. If you have poor soil, then any general-purpose product should work. There are both traditional and organic fertilizers, whichever you prefer.
As far as why the flowers fell off --- it could be the heat. Sudden changes often cause plants to react by dropping flowers or leaves. Once the plant adapts or the weather cools, your plant should start blooming again.
As far as watering goes, it's best to let the soil surface dry a bit between waterings --- you don't want it to stay wet or soggy for extended periods.
---Justin, Senior Garden Editor, BHG.com