So it is declared, that there are cruise missiles with nuclear reactors on board. As I understand, when it's working, the neutrons keep flying from off the fuel. Neutrons give induced radioactivity. Does that mean that everything around such a missile becomes not just filled with products of fission, but radioactive itself?
This idea has been around for a while. The Aircraft Nuclear Propulsion (ANP) program in the 1950s was meant to test this possibility. A cruise missile is essentially a pilotless aircraft. It employed a liquid core nuclear reactor that exchanged heat with air brought in by a turbofan to operate as a type of jet engine. A B-36 bomber was outfitted with a nuclear reactor in its bomb bay to test the feasibility of operating a nuclear reactor on board an aircraft.
The program never materialized much. In fact there is a claim that this nuclear B-36 dropped the reactor because of a mishap, though this is was officially denied. Often when something is officially denied it becomes actually more plausible. The program was axed by President Kennedy shortly after taking office as being a money drain that produced little foreseeable results.
Premier Putin of Russia now boasts of having a cruise missile that is capable to flying anywhere to deliver a nuclear bomb. While I suspect there is a program in their works to develop this, I doubt this is a working system. Getting a nuclear powered aircraft to work is a difficult project and I suspect they have not so quickly produced this in working form. Putin also states they have a sort of unmanned submarine or large torpedo that can travel around the oceans to deliver nuclear bombs. I think that is more likely real, and a bit worrisome.
Neutron shielding can have reasonable thickness (say, 30 cm) (http://oa.upm.es/4596/1/INVE_MEM_2008_61569.pdf ). However, if nuclear weapons are used, we will have greater problems to worry about...
The characteristic length a neutron would travel in air before being absorbed is about a dozen meters. That is at sea level, at high altitude the path would be larger but neutrons would not reach the surface. Absorption cross section is much higher for nitrogen, than for oxygen, so that neutron would most likely be absorbed by a nitrogen-14 nucleus, producing stable nitrogen-15 isotope, and so no additional radioactivity would be induced.
I do not know the specifics of Putin's missile design, but in all designs of nuclear-powered rockets from the 70-s or 80-s that were intended for use in atmosphere, nuclear fuel and fission products do not leak into air. So if all you are worried about is additional radioactivity from a missile flyby you are in luck.