Invert Y-Axis Water Spray in Super Mario Sunshine

Sadly, there is no way to achieve what you are proposing. The game settings itself do not give you the opportunity to change the axis' of movement while spraying and changing the game files itself would be illegal as you would be changing Nintendo's intellectual properties.

On a different note, I am certain you are playing using a legal copy you obtained by ripping your own game disc to an iso.

It may be possible to do with an action replay. However, if you are using an emulator (presumable Dolphin) you can achieve this using the following steps:

If you are using a Gamecube controller, make sure your adapter is set to PC instead of Wii U.

Right click on Controller Stick Up, and add the following:
([your input for backward] & !([your input for R])) | ([Your input for forward] & [your input for R])

Then, right click on Controller Stick Down, and add the following:
([your input for forward] & !([your input for R])) | ([Your input for backward] & [your input for R])

You can invert Up and Down only when R is fully pressed. This will allow you to spray while running by half pressing the R trigger, just like the original Gamecube controller.

An example (full press threshold = 90%):

Up:

(`Left Y+` & (`Trigger R` < 0.9)) | (`Left Y-` & (`Trigger R` > 0.9))

Down:

(`Left Y-` & (`Trigger R` < 0.9)) | (`Left Y+` & (`Trigger R` > 0.9))