The astronaut couldn't find out where he is or when, if he doesn't have data from the night sky on earth. He would need more than a Polaroid of the sky at night^^. I guess NASA won't expect that so they wouldn't put such data into space ships.
But if they have a data link to earth (Think Stargate, where radio waves could go both ways, matter only one way.) and a strong enough telescope at their end of the portal, they could: Find a stars, a galaxy or a black holes that are the same distance from them and from earth. The light going out from them would reach the earth and the portal exit at the same time. If they find one, it would be easy to get your position relative to Earth by looking at how it is rotated. For example if you see the galaxy from earth at a 45° rotation. And from the portal at 135° you had a triangle with a right angle (a²+b²=c² and a=b). If they find more than one it would be even better. 99
If they don't find anything at exactly the same distance with earth. Maybe something recognizable even if it is 2 billion years wrong. And than you could find the Milky Way and look for our sun. (There shouldn't be anything to see, since light of our sun has only travelled 4,5 billion light years, cause our sun is 4,5billion years old. But we should be able to see his sun, if it is older than 6 billion years.) Or some other light source that was old enough.
If the wormhole also travels through time*, it would still be possible to find something that has send the light at the same time out, and reached the astronaut and earth at roughly the same time. And you could find out that happened, because the calculated distance of the wormhole exit and earth wouldn't match. (You would need to look at something near us that is old enough that light reached the astronaut.)
*Well as long as the wormhole exit has light from stars come to it that also travelled to earth. check out https://www.space.com/33005-where-is-the-universes-edge-op-ed.html So it would be impossible to find out for the astronaut, if the wormhole would send him over 26 billion light years away (without time travel).
So I would say it is possible, but if he is alone with no communication, he would need years, strong computers and a good (radio) telescope. NASA would probably need also years just to get the necessary equipment through the wormhole. 99