As the name implies, precalculus covers a number of important topics that will come up in calculus, including a lot of trigonometry and concepts from algebra. Ideally, you should have a firm grasp of this before starting calculus. Of course, you should also have a firm grasp of basic algebra before starting calculus. Have you looked into taking a placement test, or looked at old syllabi from calculus courses taught at your school to see on your own if you’re really as unprepared as you think?
You’ll want to take calculus-based physics if you plan on studying engineering. That means you’ll already have taken or will concurrently take calculus when you’re taking physics. Algebra-based (non-calculus) physics is “physics lite” and is virtually useless for anybody who’s planning to major in engineering or the physical sciences, as you can’t properly understand most concepts in physics without calculus. Calculus-based physics is like learning about the recipe and ingredients needed to bake a cake on your own. Algebra-based physics is like going to the store and simply buying a cake.