That is completely expected… RGB are screen colours, the colour spectrum is larger and so what you see on your screen is alway brighter.
Trouble is, even if you work in CMYK, the colours you see on the screen will be more vivid, and unless you have an accurately calibrated screen, there will be some change once printed.
One thing you could do is get CMYK printed swatches, so you can see what for example a Blue 70-15-20-0 looks like once printed, and adapt your artwork.
What kind of file are you working with?