How does the shop work?

Solution 1:

As you already suspected, you're improving the stat a little, upgrade by upgrade. Each upgrade level of a stat costs more than the previous level.

There are no one-time upgrades that go from inactive to active when you fill the bar. The Wideband shot might seem like that, but what you're actually buying is increased frequency of that shot (and possibly, potency, but I can't really tell).

Try paying attention to the projectiles you shoot during the initial lull to observe those upgrades in action. They don't really do that much at all, at least initially.

Unspent star bits are saved, of course.