Correcting your sentence: Neither "fly on a jet pack" nor "fly with a jet pack" sounds idiomatic, do they?
Ronald Sole gave you two very good alternatives.
Flying "on a jet pack" makes it sound as if you are sitting or lying down on something. Most jet packs I've ever seen (very few) are strapped around the person. Now if we were talking about a flying carpet, "on" would be fine. 😊 Of course, if they invent a jet pack that one lies or sits on while flying, your sentence would be fine.
Your second example, "I fly with a jet pack above the lake to attract tourists," is acceptable. The word "with" can be used to show instrumentality. In other words, you are using something as a tool. For example, I am driving nails into the wood with a hammer.