Airbus can't really move a large distance so finding a landed ship is pretty hard, so best bet is to get a new ship that can equip weapons.
The problem is, I can't find any aircraft armaments.
Try shadowing with faster vessels and/or buying the Aircar if you come across a light gun for it somewhere (2x25mm or better).
Where can I obtain aircraft guns aside from looting the aircrafts? I read Air Strategy Guide or something (found that at ufopaedia on the Internet) and it said that aircraft weapons can't be bought.
...oh, wait, and are you sure you know how to actually make time pass when an air combat is set to 'shadow'? Click the minimize icon in the upper left of the combat window! It'll collapse the window and actually let the Geoscape clock advance and the enemy fly around/continue to its destination. Same as in vanilla X-Com.
Yep, that's how I tried that.