Solution using Modulo

A simple solution that catches all cases.

degrees = (degrees + 360) % 360;  // +360 for implementations where mod returns negative numbers

Explanation

Positive: 1 to 180

If you mod any positive number between 1 and 180 by 360, you will get the exact same number you put in. Mod here just ensures these positive numbers are returned as the same value.

Negative: -180 to -1

Using mod here will return values in the range of 180 and 359 degrees.

Special cases: 0 and 360

Using mod means that 0 is returned, making this a safe 0-359 degrees solution.

You're passing in the arguments in the wrong order. std::atan2 expects the arguments in the order y,x, not x,y.

Yes, that is incredibly dumb, but it's related to how the tangent function is defined in the first place (which is defined as a ratio of the y-component to the x-component, not the other way around), and like many notational mistakes in mathematics, inertia set in thousands of years ago and you can't push back against it without being a crank.

So write your code like this:

float radians = atan2(1, 0);

Or, if you want everything as explicit as possible:

float x = 0, y = 1;
float radians = atan2(y, x); //Yes, that's the correct order, don't @ me

And you'll get the results you expect.

Your second problem is that the values that atan2 correspond to don't match up with the directions you want. What you want is a circle where the top is 0°, the right side is 90°, the bottom is 180°, and the left side is 270°. Punching the values into atan2 is instead going to produce values where the right side is 0°, up is 90°, left is 180°, and down is 270°.

^{Also, in comparing with my own hardware, my y-axis is flipped compared to yours. Mine is y+↑, whereas your setup appears to be y-↑}

So if you want to transform the normal atan2 rotation into the rotation you want, you'll need to transform it like this:

float radians = atan2(yaxisval, xaxisval);
float degrees = (radians * 180 / PI);
degrees = 90 - degrees;
if(degrees < 0)
    degrees += 360;

Then, all you have to do from there is possibly transform the y-axis depending on whether you expect the joystick pushed up to return a positive or negative value. That's up to the domain of your program.