The easiest way to convert my image to 300 dpi and CMYK with Photoshop

Ask the printer. Printing is not a generic process.

Is this printing in black and white on a 4 colour page, or is it a black and white print run using only black ink? This is extremely important too - if you use a profile meant for colour printing on a black only page, you will get washed out greys.

I'm on photoshop right now and going to save the image to CMYK. Which profile should i use for generic black colour printing? (Based in US/hongkong)

No it isn't at 300 ppi (dpi). its 72 ppi. But that's cool. If you need to change it use Image Size as I said: no resample.

No, you did something very wrong. You increased the resolution (we can tell because you have more pixels). Do NOT do that! In image size, turn off resampling and you will change the ppi only Without affecting image quality.

Your original size, 3.94 inches, was exactly 100mm, so it was perfect before you changed it. GO BACK TO THE UNEDITED VERSION, don't reduce the file you increased.

The resolution and the size are set separately. It's simple arithmetic though.

300 ppi (same as dpi) is 300 pixels per inch.

1181 pixels, with 300 pixels in each inch, means there is 1181 / 300 inches. That is 3.94 inches, both width and height.

So, if your image is to print printed at just under 4 inches, it is the perfect number of pixels, and you can change it to 300 ppi. If you want to print it bigger than this, your image is not suitable. If you want to print it smaller, we can make it work.

I dont know will it help you or not but you can try this website Online RGB to CMYK conversion  for the conversion.

First ask the printer which specific CMYK profile to use. There is no such thing as a generic CMYK. Then Edit > Convert to Profile.

A quicker way is Image > Mode - but note that this directly converts to whatever you have as Working CMYK under Color Settings.

Photoshop out of the box comes with US Web Coated (SWOP) as working CMYK. This may well not be the right one, especially if you're outside North/South America. Then it's guaranteed to be wrong.

Resolution is given as ppi (not dpi). It means pixels per inch, and this simple calculation determines a physical print size for the pixels you have available in the file. Put another way, it determines how big each pixel will print on paper and thus how much the pixels are spread out.

So what you do is open Image  > Image Size. Keep "resample image" unchecked !, you don't want your file changed. Put 300 in the resolution field. Depending on how many pixels you have in the file, a physical print size is calculated from that. If you have enough pixels in the file, the size will be at or above the required print size. The you can just click OK, and this will be added to the file's metadata and the file printed accordingly.

If you don't, it's back to the drawing board.