You do what needs to be done to make your files look good online too! There's no rules about this. Sometimes, the colors could be more saturated online to make them more vibrant and it's true that the real RGB black will look more black then the CMYK one that can look grayish on the screen.
Some tools you can use are "levels", or "selective color" adjustments, or the "curves" or even the "replace colors" in Photoshop. There's a few other ways but these tools can help you to replace a color you don't like or to adjust the contrast/darkness/brightness if necessary without affecting too much (or at all) the other colors. But that also depends what needs to be modified and your knowledge about these tools. You'll need to make some tests and read about it.
You can use the brightness/contrast and saturation to also boost a bit your colors. There's also a nice curve your can use in the curves to achieve this very quickly. You'll need to make your own tests.
If you need to always replace the same kind of color (eg. the CMYK light black to a RGB 0-0-0 black or similar), you can always create an action and use it on all your files OR it's also possible for some of these tools to save an adjustment and import it in another of your file.
There's no automatic process that is bulletproof but you can still create your own shortcuts.
You can freely use a version for your online needs and one for printing.
Example for the "Levels" tool:
how to compensate 50% opacity white over photograph