One thing they could do is to ask questions about his home worlds with the intention to find inconsistencies that prove he's making things up. That would as a side effect also make the reader familiar with that home world.
"So you are from a far-away planet? So where is this planet?"
"Well, do you see that bright star over there?"
"You mean Vega?"
"I think you call it Vega, yes. My home world is a planet orbiting it."
"But isn't Vega a young star? A planet around that would certainly not have had enough time to develop life!"
"You're right, it never developed life. It's not our planet of origin. We settled on it a long time ago, however, and now you wouldn't be able to distinguish it from a planet with native life."
"So, describe some animal that lives on that planet."
"Well, there are some pretty birds. [description] And there are those nasty Bullitows, which are a real plague."
"But if you brought life there, surely you would made that planet a paradise!"
"Well, not quite. You know, there are always some unwanted species hiding somewhere in the space ships. You humans always consider space ships as clean, essentially sterile environments, but that couldn't be farther from truth."
"So you claim you've got the technology to travel through space at tremendously high speed, but don't have the technology to keep your ships clean?"
"It's simply impossible to keep unwanted life forms completely away from bigger space ships. If you had ever been on one, it would be obvious to you."
"So how come that you didn't bring any unwanted species down to earth when you came here?"
"Well, I didn't come with a big settler's ship — I obviously didn't come to make this planet a colony of ours —, but with a small personal space ship. Those are much easier to keep unwanted life forms off. But honestly, I cannot completely exclude the possibility that I've brought some."
"But if you had, we certainly would have heard something about that by now, right?"
"Not necessarily. Some species need centuries until they even reach a noticeable population. Also, how do you know that the insect that bit you yesterday wasn't actually an alien life form?"
Another type of question could be why it is so non-obvious that he's not human. Why does he not look alien? How did he learn the human language so well? How could he adapt to human culture so well that he wouldn't stand out? Why can he eat the same food that humans eat (or if he doesn't, why did nobody notice, and where does he get his food from)?
And of course the question of how he could come to earth. Of course they would question the ability of travelling fast enough, but that one he can easily dismiss with "I don't know how it works, I've never been into science; I just know that it works." But then, there would be questions like how he could land on earth without anyone noticing, despite the military watching every section of the planet.
Oh, and BTW, it would be unrealistic that "show us your space ship" never gets asked. You might find a good reason why he can't (maybe his landing didn't go well, and it sank into the deep ocean?).