Can a country take another one to court for something it did to it in the 1800s?

Sort of, but who is going to enforce it?

There is a dispute between the UK and Argentina over the ownership of the Falkland Islands AKA Islas Malvinas, dating back to the 1700s. As I recall, the issue is under very slow negotiation in the UN.  The UK has in fact ruled the Islands all this time.  

It depends what you are referring to and I doubt if there is a limitation on things like war crimes. Nazis are still being chased for WW2 crimes.

If one country took another country's statues or art, there might be a case for demanding their return. 

But for one country to demand restitution for the crusades or something would be illogical and a bit too late. 

Depends what it was, many things have a statute of limitations.