I think Dr. Kirkish says that they assumed, based on depth, that the ship had been there for about 1000 years. If you are familiar with (and really understand) the stratigraphy of the location, you could indeed make a pretty good guess at the time of deposit of an object you have not yet excavated, based on how deep below the surface it is. But, that really only applies to objects that were at surface level at time of deposit, and then became buried over time. If the ship was buried on purpose, it would intrude into much older layers. This would become apparent during excavation, but it doesn't seem like something their "sonic probe" would be able to pick up on.
On the other hand, I think she may have said "AT LEAST 1000 years", which could be accurate if they had a good date on a layer on top of the ship, but nothing else. It would be pretty meaningless if you couldn't guess at just much more than 1000 years, considering how big a window that leaves.
According to the Wikia, the ship was at 300 feet below the surface, which is "only" about 91 m (you have an excellent memory, Springer, even if your brain substituted meters for feet, which is an entirely reasonable thing to do ... I refuse to believe that archaeologists decided, sometime between the 21st and 23rd century, to abandon the metric system). At first glance, that, to me, seems like a completely ridiculous depth for a 1000-year-old something. But then, my ideas of what is or isn't reasonable are entirely based on my experience on Earth. I really have NO idea what would be reasonable in a volcanic region on Mars. The other thing about that depth, though, is that those ships are really big. In the images we see of the excavation, it looks like some of its spidery legs are practically at surface level. I suppose we can't see if they took off a mound from the top ...
I too laughed at how the Shadow ship shows up and just shoots the other ship out of the ground. "Screw your careful excavation!!". Couldn't they have shown up a bit sooner and saved the archaeologists all that unnecessary work? Although, apparently these 23rd century archaeologists have some spiffy methods. Dr. Kirkish says "it took us weeks just to excavate half of it" like that's an unusually long time. Those ships are BIG! I guess perhaps they also had a pretty large crew working on that ...
"What were they doing there?" is indeed an excellent question. If the 1000 year date is indeed correct, it could be something war-related, but it could also be that the Shadows had some interest in being near this young race that would soon be getting ready to go into space (assuming a 1000 years doesn't mean that much in Shadow time, with them being as old as they are).
[OT: I'm sitting here looking at my post before I hit "submit", thinking "Oh my god, what have I become??" All of this, about something like five minutes worth of TV from 20 years ago? But, anyone reading along clearly isn't that different from me
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