How do I connect or secure this 2.22 cm water filter pipe to my 2 cm faucet spigot?
First off you will have to unscrew the end (#1 as you have labeled) it from the faucet riser pipe. This part that you are removing is commonly called the aerator. Then you will have exposed male threads on the faucet pipe. If you are extremely lucky you may find that your water filter with the female threads will screw right onto the end of the faucet pipe. 

But be aware that there are a plethora of sizes faucet pipe threadings for aerator attachments. This type of size gauge (which you may find hanging on the faucet parts aisle of some hardware stores and home stores) can give you an idea of just how many common sizes there are available.

 

Note that you can indeed sometimes find special adapters that convert from one size of aerator thread to another. A specialty plumbing store may be able to help but I would suggest that you start with your water filter manufacturer or supplier first.