The meaning of these two sentences is slightly different, and you probably mean the plural version.
In the plural version ("any questions [...] answer them"), the meaning is that there will be a period of questions-and-answers, for some amount of time and you will answer whatever you can. This is pretty common, so I suspect this is what you mean. If so, use this version.
In the singular version, the literal meaning is that you will allow one single question to be asked, and then you will answer that one single question. After that, you make no promises about answering more questions - and if this is really what you mean, then I'd expect that you would stop answering after the first. This is an unusual thing to say, so if this is what you mean (especially as a non-native speaker), you should emphasize the fact that you do mean a single question! e.g., if you said "ask me any one question [...]", with the verbal emphasis, it would be understood this way. Otherwise, people might think that you actually meant the plural version, and simply made a mistake.