Improve similarity between rgb and cymk color mode

try exporting the rgb version as a png then place it in your cmyk file and use the eyedropper tool to try to match it

Those glowing, almost flourescent, colors are not reproducable in CYMK. Once you have a CMYK file, you can tweek the colors to get more contrast, but they won't match the RGB exactly.

You will get a closer match to RGB using spot colors instead of CMYK equivalents. You'll find them in the Swatches panel menu > Open Swatch Library > Color Books.

Unfortunately that is also what I get when converting your RGB image to CMYK. 

There is no other way than to modify the result manually.

When you know that you need to print it, start in CMYK mode.

On top of setting up your color management correctly. That blueish green at the bottom of your logo is just too bright to be printable.

Some colors simply cannot be printed with a regular CMYK process. But in your case it's not even that. You need to start with the fundamentals and use actual color management. The shift to yellow clearly indicates that this isn't just a gamut or density problem, it basically tells us that you are not using any real CM. For print stuff CM is inevitable and mandatory or you never will be able to predict how the print output will look. I suggest you read the online help on some basics:

https://helpx.adobe.com/photoshop/using/understanding-color-management.html

Anything outside that would require info about your CM settings, source and target profiles, moniztor profiles etc., but as per the above if you're not using any CM it becomes sort of a circular argument that goes nowhere.

Mylenium