Detecting air flow in a room
In the room I stayed I had a problem with air flow. Even when door and windows closed, a little bit of air still flows


This is normal. All residential rooms need some airflow. A completely sealed room will have severe moisture and mold problems. Plus you will be dead by morning on account of breathing all the available oxygen.


  and that make me sick especially on winters.


No, it didn't. You got sick because you were exposed to a virus somewhere. Probably an elevator.


  So I want to detect this air flow with a cheap system. One of the ideas that come to my mind is using Infrared Thermometer and make an image of the room. The places where air flows should be cooler then other sides.


Maybe. IR thermometers will detect several-degree differences and usually not from air. 

For a cheap system, get some tape and a roll of toilet paper. Attach 50cm strips to the ceiling and watch it move.

If you determine that you have a strong draft ("airflow" is normal, "draft" is unwanted air movement) from a window, have the building owner repair the seals. This is an easy and cheap job. If the draft is coming from corners or gaps between the wall panels your only option is to move.