How to examine person's awareness of a certain topic? I am conducting a case study on the topic of refugees and asylum seekers. My particular interest is: How aware is my test subject of the reasons (political persecution, famine, war, ...) why these people are leaving their home country?

I would like to avoid asking explicit questions like "How often do you think about the political situation in Syria?"

So, is there any standardized or way of investigating awareness (not knowledge) of a certain topic?
I think what you may be after is the accessibility of people's attitudes. The idea is that people not only have different attitudes, but that they also differ whether these attitudes are on top of their minds (or how easy it is for them to retrieve them, when they are prompted to indicate an attitude). If an attitude is subjectively important or often thought about, it should be highly accessible. On the other hand, for other things, people may have a very negative or positive attitude, that is, nevertheless, relatively inaccessible to them. Accessible attitudes have been shown to guide behavior more strongly (there is a higher correspondence between attitudes and behavior). The classic paper is by Fazio and colleagues (1989).

Attitude accessibility can be measured by measuring the reaction time to attitudes questions. If people are faster in answering an attitude question, this is taken as an indicator of a highly accessible attitude. This would also be the kind of indirect measure that you might have in mind.

So in your example, you could ask participants to agree or disagree whether certain reasons for seeking asylum are important, and measure the reaction time of their answers. Quicker answers would indicate that these reasons are more accessible to them. 

References

Fazio, R. H., Powell, M. C., &amp; Williams, C. J. (1989). The role of attitude accessibility in the attitude-to-behavior process. Journal of consumer research, 280–288.